A Monk's Musical Musings

Freestyle Convertible Counterpoint, Part 2

2012-01-23T22:27:00.000-06:00

In the first post in this series, I demonstrated a method for finding all possible vertical-only and horizontal-only shifts at the thesis points for a five-measure fugue subject and its tonal answer, and in all four quadrant orientations (Original, inverted, retrograde, and inverted retrograde). As I intimated then, this is much more practical than Taneiev's theoretical treatise, it has the advantage of not relying on mathematical formulas, and it also has applicability beyond the strict style. Additionally, the method takes advantage of the copy-and-paste and playback abilities of Encore, so it's far more useful for the composer. I can't stress too much that this method goes beyond fugal composition to any type of contrapuntal writing: Sonata themes can be subjected to the same treatment.

Today, we will continue the process for combination shifts. This will use the same method as for the horizontal-only shifts, only now at all of the intervals other than unison/octave. Just as I went through the most distant canons to the closest canons previously, I'll do the same thing now working logically through the vertical shifts in order:

V= +2/(-7), H= +4/-1, +3/-2, +2/-3, +1/-4
V= +3/(-6), H= +4/-1, +3/-2, +2/-3, +1/-4
V= +4/(-5), H= +4/-1, +3/-2, +2/-3, +1/-4
V= +5/(-4), H= +4/-1, +3/-2, +2/-3, +1/-4
V= +6/(-3), H= +4/-1, +3/-2, +2/-3, +1/-4
V= +7/(-2), H= +4/-1, +3/-2, +2/-3, +1/-4

The top line would read, then, Vertical shift of a second up, or a seventh down, with Horizontal shifts of plus four/minus one measure, plus three/minus two measures, plus two/minus three measures, and plus one/minus four measures: Simple process of elimination. Repeat this with thirds, fourths, fifths, sixths, and sevenths, and you're done.

Here's how it worked out with my, "super subject." I chose this subject to develop this process with, by the way, because it is a best-case scenario with respect to all of the combinations it makes (Or, it's a worst-case scenario with respect to the amount of labor involved! lol).

None of the plus second/minus seventh combinations worked in the first subject-subject step, so we start with plus third/minus sixth, and one measure of overlap.

Note first that here in minor, the +3/(-6) shift is to the relative major.

Note secondly that, since the subject makes a four-voice canon at the octave, that these +3/(-6) shifts could be used to double the subject in thirds or tenths above, or sixths or thirteenths below. This will be a nice orchestration possibility when I get to a fugue of that magnitude.

The closest +3/(-6) shift did not work, so we're on to the +4/(-5) shifts.

And that was the only one of those that panned out, so here we are at +5/(-4) shifts.

Note that +5/(-4) shifts are at the answer's level, and we already know that two of them work, so this is really no biggie.

This +6/(-3) shift is to the relative minor from the major mode perspective, so that's kind of interesting.

You'd have to double the leading tone to get this one to work out, but with three or more voices and an irregular resolution, it is possible. In fact, one of the things that makes freestyle counterpoint generate so many more viable possibilities than the strict style, is that you can make a lot of combinations work out with creative use of harmony.

Here we are at +7/(-2) then.

This offers some interesting modal and modulation possibilities, and that finishes up the subject-subject combination shifts.

Since there are really only two intervals different between the subject and answer, this will seem somewhat like a rehash. Again, none of the +2/(-7) shifts work.

These do have a different flavor and function though.

And again, all but the closest possible arsis point works. Oh, by the way: Pay no attention to the page numbers. I had to break the Encore files up into smaller chunks after the fact to get them to convert to PDF reliably. Not sure what that bug is, but beyond about thirty pages the conversions are not sized properly.

The +5/(-4) shifts put the answer form on the tonic level, which is unusual, but only this one measure overlap works.

And on to the +5/(-4) shifts.

And these are quite interesting. A composer would likely never stumble upon all of these through casual experimentation, which is the point of this exercise.

And the +6/(-3) shifts.

From the minor, this combination provides a modulation from the dominant directly to the relative major. Very hip.

Now the +7/(-2) shifts.

The way this has worked out, I'll write modulating thematic phrases with the combination shifts that overlap for two measures. These are to be highly prized because it's much easier to write a modulating episode or interlude than it is to modulate with strictly thematic material.

And now for the subject-answer combinations. Here - and in the upcoming answer-subject combinations - you have to keep in mind that the subject and answer are already shifted in relation to one another, so here the answer is a step higher than it already was, answering on the sixth instead of the fifth.

More unusual modulation possibilities.

Here's the first instance where one measure of overlap doesn't work, but two does. Some remote modulation possibilities here.

These combinations are actually like the original canon, but with the answer form following on the tonic level. No big deal, IOW.

Yadda, yadda, yadda/blah, blah, blah. lol.

A perpetually modulating canon could be made with this type of arrangement: Up a step every iteration.

These are so close to the +3/(-6) subject-subject combinations that the difference is trivial.

Still nice modulation possibilities though, and that ends the subject-answer possibilities.

And the last thing to cover in Part 2 are the answer-subject combinations.

Still, some nice subtle variations on the modulation possibilities.

But plenty of trivialities too.

Obviously, very similar to the same subject-subject and subject-answer combinations. I'll group them and compare them later.

With most subjects and answers - those not composed from the outset as canons, like this one was - the composer is looking for whatever works. With the embarrassment of riches here, I need to figure out what to discard.

I'll trash the single measure overlaps, use the two measure overlaps for modulations, and the three measure overlaps - there are only a couple so far - I'll put in places of special prominence.

This rocking combo enters on a v(m7) sonority in the minor and a V(m7) chord in major. Won't find those in Palestrina. lol.

Obviously, I could have skipped all of these trivial one measure overlaps if I weren't doing this as a demonstration, and just hunted down the best stuff.

This is a cool combination. After the answer on the dominant level, you get the subject on the dominant level, which confirms the modulation. Precede this with a subject statement in the tonic, and you'll have some nice modulating stretto possibilities.

Because I originally wrote this as a canon, this one works too.

This is the only non-original subject-subject combination that works with four measures of overlap, but it's not really a big deal because only the head figures differ between the subject and answer. Still, I'm glad I made myself think of this angle.

Another modulation candidate.

And one last unusual combination with three measures of overlap.

I'm keeping the systems large and not worrying about the clashes because it's easier to enter the notes on the largest size staves. When I get to the audition and construction phase, I'll reduce their size and clean them up.

*****

So then, here are all of the combination shifts at the thesis points to add to the vertical-only and horizontal-only results. Next step is to do the horizontal-only and combination shifts for the arsis points. Off-beat entrances can be very sublime, so I'm hoping a few nice ones turn up, but I'm also hoping for less than forty-four pages of results! lol.

Freestyle Convertible Counterpoint, Part 1

2012-01-17T00:27:00.000-06:00

I first read Sergi Taneiev's treatise, Convertible Counterpoint in the Strict Style, in 1987, I believe. It was a seminal moment for me, as it opened my mind up to the staggering possibilities of vertical, horizontal, and combination shifts with respect to contrapuntal combinations. I couldn't wrap my brain around the formulas, though, because like the overwhelming majority of musically minded people, I don't have any talent for math.

There is a reason for this, by the way: The musical mind works in terms of sound and sight - this is why composers need to see the music on paper - and numbers are mute and invisible: If I can't hear it and see it in my mind's ear and eye, it is not possible for me to manipulate it in any way. As I said, this is the case for most musicians, with Taneiev being a spectacular exception.

Nonetheless, I attempted to go through the treatise again back in 2005 when I started this weblog - links in the sidebar - but I abandoned the project when I realized that the solution for me - and those musicians like me - was to simply do the calculations mechanically (While my math aptitude scores are at a dismal 42nd percentile, my abstract reasoning and mechanical reasoning scores are in the top one percent, so I can rock with this approach).

The other problem with Convertible Counterpoint is, of course, that it only applies to the strict style. Unless you want to sound like Palestrina then, it's truly useless, even if you can wrap your brain around the formulas.

Once I had the idea for a mechanical approach, the only thing left to do was to develop the methodology, and of course that was a monumental organizational challenge, and composers tend to be better at organizing things than most. So, here we are.

What I did was to break the process down into tasks that were logically ordered, and that covered the most valuable and easy to employ combinations first, and saved the more esoteric devices for later. The process itself establishes a pattern at the outset with the more obvious possibilities, and that pattern is repeated with the latter and more remote concepts. Also, I organized things to take advantage of Encore's copy-and-paste abilities, which makes this monumental task quite quick and painless.

Without further ado then.

Step 1 is to write out the subject and answer - or countersubject, or counter-answer, or sonata theme, &c. - in both modes and in all four orientations. Note that with tonal answers on the dominant level, inversion makes them go to the subdominant. I shall exploit this. lol.

Some may wish to omit the retrograde forms, but I find that working with them is great mind exercise.

Simplest of all combinations are the pro forma vertical-only shifts, because there are only two possibilities, and they always work: Lines can be doubled at thirds above/sixths below, or vice versa.

To take advantage of Encore's copy-and-paste capabilities, you should always make the thematic elements Voice 1, and the shifted elements Voice 2, whether the shifted elements are above or below. That way, you can just solo Voice 1 and copy the four versions for the next operation. Needless to say, a massive time-saver which makes this epic project much more bearable.

This now explains itself.

After the vertical-only shifts have been done, the next step is the horizontal-only shifts, and not just any horizontal-only shifts, but those that begin on the strong beat (The beginning of the measure in 2/2 here). Those are called the thesis points, versus those starting in the middle of the measure, which are the arsis points (Yes, I'm aware those terms have been reversed in the past, but this is the original - and technically correct - way these terms were employed).

Since I composed this subject as a four-voice canon, all of the thesis points work, and there are two ways of looking at the shift: How many measures from the beginning, and how many measures from the end (Or, how many measures overlap). With a five measure subject, the absolute value will always equal five.

And again, things become self explanatory.

I also note potential traps, as I've done here. The minor version of the canon uses unequal parallel fourths and fifths to maintain invertibility, but in the major mode the fourths are both perfect, so inverting them would give parallel fifths.

And so here is the closest canon. Of course, you can also solo the parts and listen to the combinations.

The tonal answer also works at every thesis point, but with caveats.

What you have to do at this point is to disregard any conflicts of mode, key, or accidentals - those things can be resolved when you get ready to employ the combinations - and just look at what I call the absolute value of the counterpoint. Some of these combinations may have to be, "modalized" to work, and those modal resultants can be spectacularly colorful (They can also suck royally. lol). In any event, don't get bogged down with any consideration that you don't have to.

Things again become obvious.

When different elements are combined, the possibilities usually drop dramatically. In fact, this fugue subject and answer are "insanely great" as Steve would have said, lending themselves, as they do, to so very many combinations. But, for subject-answer combinations here, only two can be made to work.

And for the answer-subject combinations, only one will work.

*****

So, here are all of the possibilities for vertical-only and horizontal-only shifts at the thesis points for this subject and its tonal answer. At this point there is a choice as to how to proceed: One can either repeat the process for horizontal-only shifts at the arsis points - which would certainly yield very limited results, if any - or proceed to combination shifts at the thesis points. To me, it seems more logical to go ahead with the combination shifts at the thesis points, as that will finish those up entirely, and then go on to the horizontal-only and combination arsis shifts, so that they will also be grouped together.

Part 2 then will be the combination vertical-horizontal shifts.

Fugal Science, Volume 1, Number 4

2012-01-11T21:27:00.000-06:00

You will have to read the first three installments in this series to get much out of this post.

To review, we started out with a fugue at the octave (Two-part invention format), turned that into a fugue with a tonal answer at the fifth, and then introduced some character to finish it as a piece of music. That's as far as I could take it in two voices with the elements I limited myself to.

So, this is the initial three-voice fugue, and I had to open it up an octave to make things work, so it's for string trio.

Here's the M4A audio: Fugue Number 4

I had to open the exposition up an octave, otherwise the answer and counter-answer would have converged on a unison. This worked fine in the four-voice string quartet version of the fugue I wrote first, but it didn't float my boat in only three. As a result, the answer and counter-answer begin a twelfth apart, just as in the guitar versions.

For the three-voice subject statement, I had to create a new element, which is a second countersubject. Many of the old school fugal theorists didn't like third or fourth lines that had so much the character of harmonic fill-ins - I'm thinking of Andre Gedalge - but if you do them right, they can have plenty enough character as independent lines. And, when you have an interesting and active primary countersubject, they are often the only kind you can write. This second countersubject is so distinctive that it would actually make a very good fugue subject. And yes, that gives me some ideas going forward.

Since this is just the initial three-voice version, I again have taken the approach of changing as little as possible to bring it into perfect balance. That means that the plan is arranged around the idea of presenting all of the possible two-voice duets in a logical order. As a result, the piece is merely competent and not killer. It does contain some killer new elements, however.

Once the viola part rests after 16 - first in/first out - episode one proceeds with the outer voices and is just like all of the first episodes that have come before.

I've probably read 99.9% of all of the old treatises and textbooks on counterpoint and fugue that have been translated into English over the years, and almost invariably they take the position that three-voice fugues offer the best combination of harmonic fullness and lineal independence. I, however, find them difficult and lacking: Four voices is easiest for me, which is why that's usually where I start.

A great example of the difficulty and awkwardness that can crop up in three voices is the transition from the end of the first episode to the first middle entries here: The only way you can change the thematic entrance order in three voices is to begin a statement in a voice that is already active, versus a more effective entrance from a voice that is resting. At least this one comes after the bass has had a whole note, which is the next best thing to following a period of rests. I had to do this, because without this particular arrangement - the outer voices - the entire rest of the fugue would be impossible. I'll explain how it came to this in a minute.

Musically, by the way, this entire section is exactly as it was in the previous guitar version. However, the section stirkes the listener differently now, as the countersubject is not a new element (It was in the exposition).

A larger problem is that the second countersubject is not employed here: It would work perfectly with a truncated first statement and then a complete second statement, but that will have to await the first four-voice version (Where I'll also be able to rotate the elements between voices), or a later three-voice version that includes more structural and formal elements.

Look first at the new element that starts in forty-four: In order for the thematic entrance to be on top with the violin, this section must be a duet between the viola and cello.

A word about voice exits: If the voice has a whole note during the dominant chord before the cadence - that would be on either sol or re. - then it does not have to participate in the actual resolution. If, however, the voice is on an active tone and/or is less than a whole note, it does (This would be fa or ti.).

Here, in this MIDI/Soundfont version, the violin seems to obscure the beginning of the duet, but in a live performance the instrumentalists would fix that quite easily and tactfully.

At forty-four is the old, "interlude" that has the subject over the bass line of the primary episode, but I found that the lead voice of the old episode also works in this combination... by only changing one note. That would be the fourth note in the viola part at forty-six, which is now a D but which used to be a G. Needless to say, a wonderful discovery. I went back and changed that note in the versions without the subject too.

When you discover combinations like this post facto, oftentimes some interesting modal harmonic successions occur: In this case, there is a D minor seventh sonority in a 4/2 arrangement in forty-six, which doesn't sound particularly strange here, but just wait until I invert it.

Note first that the viola had to participate in this resolution because it is a fa-mi formula, and also the previous note was a half. Also note that the viola's first cadence was to the tonic, and this one is to the dominant (Even if it is functioning as the major third in the relative). Otherwise, the music in this duet is unchanged from the previous guitar version, with the exception of the single note in the episode mentioned earlier.

That leads to a duet between the upper voices at sixty, which must be that way because of the next section. The music itself is again unchanged from the previous version.

At sixty-nine is the second three-voice combination of the episode and the subject, and in the previous fugue this lead into the pedal section, which it also does here, only in a different manner. Now, the pedal is introduced by the cello, who just holds the last note of the subject over. During the pedal point proper, the viola has a new part that perfectly fleshes out the combination and which also increases the energy of the final half-cadence magnificently. This produces an air of invincible inevitability as the viola descends through the cadence toward what can only be the beginning of the recapitulation canon.

Note that in measure seventy-one there is a root position G minor seventh chord. Groovy, baby.

Since we have three voices to work with, we get a three voice concluding canonic stretto (I composed the subject as a four-voice canon, remember).

One of the elements that is improved over the two-voice extraction is the final hyper-stretto between the subject in the lead, and the augmented subject in the bass. First of all, the pickup figure in eighty-six is now to a non-thematic part, so the thematic entrances stand out better. Secondly, the third voice mediates effectively by more completely defining the harmonic context. Sure, the listener is going to be at least subconsciously aware that the minor-ninth-plus-octave in eighty-eight belongs to the V(m7m9) sonority, but having the minor seventh present makes things much clearer. Also, at the beginning of ninety, we get a complete augmented triad that is just awesomely bad ass.

The fun continues as the third part adds to the power of the already powerful conclusion too.

So then, the plan is: Exposition, outer voice duet, lower voice duet, three voice interlude, outer voice duet, upper voice duet, three voices from the concluding episode on. Nicely structured and balanced. How I finally arrived at this arrangement solution is a great old trick: Work from the end of the fugue back to the beginning.

I had the exposition and the hyper-stretto - alpha and omega - and the three-voice recap canon was obvious too. Then, the pedal section was obvious, and only a duet between the upper voices could lead into it. An outer voice duet was the best preparation for the upper voice duet, and the three-voice interlude could only be preceded by a lower voice duet, so that lead back to the first outer voice duet. That's when I found the exposition and first middle entries could only be joined as they are: The exposition-following episode can only be between the two outer voices.

I'm not at all convinced that this fugue will survive long without modification, but I am convinced this is the best arrangement for this limited number of elements. I'm currently working on an exhaustive analysis of all of the possible combinations this subject and countersubject can possibly make - I finally cracked a mechanical solution to Sergi Taneiev's Convertible Counterpoint formulas - and that may very well redefine these fugues all the way back to Number 1. That will be next, but it's going to be a while.

Fugal Science, Volume 1, Number 3

2012-01-10T17:27:00.000-06:00

To recap, the first version of this fugue was a fugue at the octave - what Bach called a two-part invention - and then, I made it a fugue with the answer at the fifth above - actually a twelfth in this guitar piece - by changing only the second five measures, and adding a new section in the subdominant. I had to add the new section to present a second statement of the answer. Otherwise, you would only hear the answer once in the exposition, which is an obvious imperfection (You need to read those posts first, or this won't make much sense to you).

As I stated at the conclusion of the previous post, I love the spartan and ascetic style I developed for those two fugues, but I've noticed a couple of opportunities to add some spice to the accompaniment parts (Countersubject and counter-answer). This, I believe, elevates the piece above a compositional/technical study into a genuine piece of music.

Here is the M4A audio file: Fugue Number 3

The only thing I added to the countersubject is the little lick in measure seven. Remembering that this started as a four-voice fugue for string quartet, that lick is in the counter-answer there. It adds gobs of color because - reading top to bottom - C, E, and G-sharp spell an augmented triad. This is also, due to fortunate placement, not overly difficult to execute on the guitar either.

This lick alone won't send shock waves through the fugue, because the counter-answer is only heard here in the exposition...

... but the corresponding lick I added to the countersubject will. In measure eighteen is the run I added, and it is a diminished scale - also called a 1 + 2 scale - that approaches every note in the diminished seventh arpeggio by half-step. Seven of the eight notes of that scale are in the lick, with only F-double-sharp missing. This idiom did not become, "normal" until the Romantic era - I can remember this same lick from Sergi Taneiev's Fourth Symphony off the top of my head - so it's not something you'd ever encounter in a Baroque fugue.

When you sharp so many scale degrees in a diminished scale lick like this, it's metaphorically like kicking up some dust: It takes it a while to settle, and it only settles by re-presenting the diatonic degrees. The following gradual descent of the countersubject fulfills this need perfectly, with the A-sharp altered tonic degree only being cancelled out by the A-natural in measure twenty. I absolutely adore this affect.

The new lick also makes the perfect dovetail joint here more obvious when it appears in measure twenty-two. After that point, the rest of the page is the same as it was way back in Fugue Number 1.

The already dramatic modulation to the dominant level is also enhanced by the new lick, and even the surprise entrance of the subject in measure thirty-four plays a part, by extinguishing the raised tonic degree early, like a gust of wind suddenly blowing the kicked-up dust away. After that point, the new feature does not return until the statements in the subdominant; nothing else on this page is changed.

And there it is in measure fifty-six, adding drama by kicking up some dust.

Here too the new lick aids in setting the perfect dovetail into sharper relief, and appropriately - as this is near the end - this modulating dovetail is more dramatic than the first one.

For the previous fugue, the goal was to alter as little as possible to bring the piece back into balance and perfection - and on that level the piece is successful - but since this is the third appearance of a version of the first sequential episode, I felt it needed one last feature, and that is a proper pedal point section.

I achieved that by inserting a two measure sequential pattern after measure sixty-eight. Returning to the theme of adding meaning to the music by varied repetition, the augmented sixth at the end of sixty-eight was heard once before in a different context when the piece modulated to the dominant region.

Those two measures are not an arbitrary addition either, since they briefly tonicize the dominant and subdominant levels, which are the first and final regions the fugue traverses, respectively. It's a cool device that provides a moment of resistance before the recapitulation canon, and you really, really feel that, "the end is near" because of it.

Still no changes on the final page, so it's the same as in Fugue Number 1. The three versions respectively are 73, 88 and 90 measures long. I view Fugue Number 2 as a transitional form, but 1 and three are fully formed. Number 1 will end up in some collection or other, but this one will certainly end up as a movement in my next guitar sonata. I already have a killer Scherzo to go with it.

*****

After completing this, I wrote a couple of three-voice versions for guitar duo, but the piece needed to be opened up an additional octave, so Fugue Number four is for string trio. That's next.

For the story behind Sibelius 7 in the trash, the grim details are here and here.

Why Sibelius Will Forever Suck Compared to Encore

2012-01-09T20:27:00.000-06:00

This is my Encore work environment as captured from the 23" 1920x1200 Apple Cinema HD Display that I use.

Is there really anything I have to explain? The operation of the user interface is obvious just by looking: You click on the palette that has the item you want to add, and then you click on the score to add it. Nothing could be simpler. Back in 1993, it took me less than 15 minutes to figure out how to use this interface, and I only ever used the manual for a reference guide. I needed to do that for only a few weeks.

By contrast, Sibelius 7 uses Tabs, so you have to click on the tab, then the item, and then the score: At least one more click to do every task. But it's worse than that, because what items are on which tab is not always intuitively logical.

See that long expressions palette on the left? Double-click on any button on that palette, and you can Edit the User Expression to whatever you want - this is where I create my analysis symbols &c. - and then place them on the score. After placement, you can change the cursor back to the arrow nib, and then move the placement of the item to wherever you want.

See the Arrow, Eraser, and Pencil icons at the top left of the score page? You click on those to change the function of the cursor. By contrast, Sibelius requires you to open a preference pane to change the behavior of the cursor.

Just to the right of those icons is the sequencer transport: Record, Play, and Stop. With Sibelius, you have to go to the appropriate tab first, adding a click to a click just to play. Also, Encore has a real MIDI sequencer that you can work with, while Sibelius has, "limited playback capabilities" even though it comes with three discs worth of sounds.

If I want to edit the properties of a note or notes, I can just click and drag the cursor to highlight what I want to edit, and then use the notes drop down to tie, slur, beam, flip stems, &c. The first time I tried to do this with Sibelius... the score moved. That's right, the cursor is defaulted to drag... which is a drag. Then I had to go to the aforementioned pref pane to fix it.

See at the bottom where there are shortcuts to the most common transpositions? I never did figure out how to do that in Sibelius, and Sibelius doesn't even play 8va/8vb clefs. Seriously, they are just there, "for show." So, my bass part here would be up in the cello range with Sibelius if I just imported this score as a Music XML file.

Sibelius does not use the main program status bar for anything except for File, Edit, View and Help basically. There are no equivalents of the Notes, Measures, or Score drop-downs as there are in Encore. This makes many tasks boatloads of needless extra work.

If I have tasks associated with measures, I again highlight what I want to change, and then I get an easy to understand list of operations I can perform. Note also that the dropdown menues teach you the keyboard shortcuts. Sibelius' anti-intuitive and user-hostile interface works very hard to keep everything hidden and secret. I have zero tolerance for that kind of crap.

When I need to do any score formatting, there's a drop-down for that too. I don't have to explain anything, do I? Clear as the Caribbean. See those MIDI functions at the bottom? Nothing like that in Sibelius (And, there's a MIDI tool on the tools palette at the upper left too: You can change MIDI settings at any point in any part by using that). Sibelius is a midiot.

To view or not to view? I can highlight and hide staves and control points here, and I can change from page to linear view too. Sibelius uses a slider for resizing, which is the ONE THING I liked about the program.

Since Encore uses the multiple floating palette paradigm, the Window drop-down is important: You can open as few or as many as you want, even ALL of them. I have every on I need visable, because if you click on the palette's title, it will cange to the next non-visible palette. That still tickles me, and I use the short Clefs palette at the lower left for that.

You'll also notice that the Staff Sheet is checked.

I keep the staff sheet out to the right of the score, but like all floating windows, you can put it anywhere. Clicking on a name opens an edit box, I can play, mute and solo parts/tracks here, also adjust the size of the staves and their transposition. This is what I looked in vain for in Sibelius.

You also assign MIDI channels and programs here. Again, Sibelius is a midiot.

I'm betting that this one post is enough to teach anybody how to use Encore. Meanwhile, the student of Sibelius hasn't gotten past the first tutorial.

*****

It's 2012. There is no excuse for anti-intuitive and user-hostile GUI's anymore. There is also no excuse for a music notation program not to have a decent sequencer. I actually had less trouble with Finale. At least it used some floating palettes; Sibelius only uses one for the notes.

I think a successful false advertising lawsuit could be brought against AVID for calling Sibelius, "easy to use." The fact of the matter is, Sibelius is THE MOST DIFFICULT OF ALL NOTATION PROGRAMS TO USE. And if you need a MIDI sequencer, you can't use Sibelius at all.

I think Ill put this at the end of every post for the rest of the year.

Fugal Science, Volume 1, Number 2

2012-01-09T18:27:00.000-06:00

As I mentioned at the end of the first post in this series, I could have turned that fugue at the octave - the two-part invention format - into a fugue at the twelfth above just by changing measures six through ten with a tonal answer and a counter-answer, but then the piece would be less than perfect, because you'd only hear the answer once (You'll want to read that post before this one, because I'm not going to explain all the details the same way in this post).

So, in this version, I have changed the exposition to include the tonal answer and a counter-answer, and I have added a section near the end in the subdominant region that has both a subject and and answer. If you are keeping up, you'll realize that the answer in the subdominant is on the tonic level, and I use this relationship to re-modulate back home. The philosophy here is to change as little as possible to bring the piece back to a state of balance and perfection.

Any time you change something in any piece of music - especially when you add a new element or section - you will usually lose something you like, so the criteria for whether to change or not is this: Do I gain more than I lose in this transaction? I got an emphatic yes, because I added a clever section.

Here is the M4A: Fugue Number 2

As you can easily see, I only changed the second five measures by replacing the previous subject and countersubject combination with an answer counter-answer one. Measures 1-5 and 11-16 are still exactly the same. The new tonal answer/counter-answer section adds a lot of energy to the piece, and the way the voices converge by step into the first episode is superior to the previous version.

There are no changes on this page whatsoever, but the perfect dovetail now means something different to the listener, because this is the first time you hear the countersubject.

As an aside, varied repetition adds meaning to music. This second episode makes the first episode mean more by being a varied repetition of it, and this episode means more because of the previous version: Win/win. IMO, what makes a lot of fugues tedious is that there is not enough repetition and corresponding added meaning. Just because you can compose a new episode whenever you need one doesn't mean you should: It may not even be the most effective thing to do.

No changes here either.

I only had to change measure 54 to change the modulation from the tonic to the subdominant, and I was sorry to see the cool pedal meld go, but the following section more than makes up for that trivial loss.

Starting in 55 we get the subject and countersubject on the subdominant level. At 59, however, the answer interrupts, which leads to a very cool musical structure...

... which is a second perfect dovetail joint, only this one modulates. Cool huh? The rest of the previous answer is now accompanied by the countersubject, not the counter-answer, so we get a perfect dovetail and a re-modulation back home. Well, this gives the listener, "Deja vu all over again" because this episode is exactly like the first one, only it leads into the final canon now. That is all I absolutely had to change to bring the piece back into balance and a state of perfection, but it certainly is not all I could have changed.

No changes here, of course.

*****

Personally, I love this austere, ascetic style, but most people crave a little more personality in the music - including me - so there are a few opportunities to ornament these lines that will add a bit of acerbic wit to the ascetic style, and elevate the piece above a study to a more artful level. That will be for the next installment.

My brand new copy of Sibelius 7, in the trash with the eggshells and coffee grounds, where it belongs. I have zero tolerance for abjectly idiotic user interfaces and crippled feature sets. Sibelius is exactly garbage.

Fugal Science, Volume 1, Numbers 1-3

2012-01-09T00:27:00.000-06:00

Back in 1994 when I was a Doctoral candidate at UNT, I came up with a magnificent and stately fugue subject that I composed as a four-part canon. With that subject, I composed a string quartet fugue that you can see and hear here.

That subject came out so spectacularly well, that I knew I would return to it at some point and write a series of fugal works with it. Well, back in 2010 I finally reduced it to its smallest form possible, that of a two-voice fugue at the octave (Bach called these two-part inventions, feeling they weren't worthy of the title of fugue, but they are just fugues with the answer at the octave), and that got me onto a very scientific track toward building it back up to a four voice fugue again.

As an aside, I have been preparing this post for a long, long time, and I was hoping to be able to use the Sibelius Classical Guitar sound file... but it was not to be. So, I'm still stuck with my soundfont collection. I'm so disgusted that Sibelius failed so spectacularly, that I'm planning to return to external sound modules again. Hard to believe that in 2012 there is still NO SINGLE PROGRAM THAT CAN TAKE A COMPOSITION FROM CONCEPTION TO FINISHED SCORE AND TRACK WITHIN THE COMPUTER. But, I'm sure the market for such things is small.

Anyway, here is the M4A file of Fugue Number 1 in A Minor for solo guitar.

One of the reasons that it took me so long to get back to this subject is that I had to completely invent a noble/stately fugal stile for the guitar. Bach's keyboard fugues usually built up to a constant eighth or sixteenth surface rhythm, but that's not an option for the guitar. Well, I have been reading about modal style vocal fugues, and I noticed that the rhythm is more based on quarter notes, and the rhythms breathe instead of building up to a motoric effect. By mixing the vocal and instrumental rhythmic styles - and basing things on quarter notes - I finally came up with the solution.

So, we get the five measure subject on the top system, and then the subject an octave higher combined with a countersubject on the second system. Measures eleven through sixteen are the first sequential episode, and there is no modulation.

Starting at seventeen, we get the subject again in the lower octave, but this time with the countersubject above. In 21, however, the countersubject is interrupted by a stretto entrance of the subject with a single measure of overlap. This would be a trivial stretto, except for the fact that I made a perfect dovetail joint out of it: The countersubject is unmodified at the point of interruption, and it continues in the bass with no modifications there either. I've never found anything like this in other fugues, so it may be a unique idea.

The unusual feature from measure 22 on is that the music is exactly the same as back in the exposition, which is basically unheard of in traditional fugues. What I'm doing is using repetition in different situations and varied repetition to affect the listener. Things get really strange by the time the second sequential episode comes up, because, "The Song Remains the Same." At the last possible point, the written-out trill figure interrupts to make a modulation to the dominant level. The bass line continues as it did the first time, and so there is an ever-so-brief augmented sixth on the final sixteenth note (Yes, that cross-rhythm is difficult to get just right). These kinds of things delight me, but I fear they are lost on most listeners.

Note that the exposition was ten measures, and the dovetail section was only nine. Also that the first sequential episode was six measures, but this one is only five. So, the first section of the fugue was sixteen measures, and the second is fourteen: The pace is quickening.

For the third section the piece modulates in a rather dramatic fashion to the dominant for a new, closer stretto with two measures of overlap. The rising bass line is really spooky, containing, as it does, an augmented triad rising into a diminished seventh arpeggio. At that point it has synched up with being the countersubject, but then the subject again interrupts in 34.

Note how the cumulative rhythm pauses there. I would never have done that in years past, but it does fall into the category of a, "dramatic pause." Once I discovered this, I decided to use it more. lol.

In 36 I got the opportunity to use an ascending chromatic tetrachord in the accompaniment line, and that leads into a slightly varied figure in 37. So, all of the joints are totally smooth, but it is not a strict dovetail.

In my fugal terminology, there are episodes and there are interludes. The first two that were variations are obviously episodes, and they have the trademark episodic sequencing. This third one comes in the place where I would put an interlude, because I discovered a wonderful contrapuntal combination as I was working with this material: The subject works over the bass line of the episode. Now, when I say, "works," I mean that in the strictest technical sense according to my laws of pure counterpoint. You'll have to look those up, as there is not space here.

Part of my two-voice style is the idea that, if it's OK in three or more voices, it's OK in only two. So, at the longest possible dramatic pause in 42, we get a diminished twelfth approached by parallel motion from a perfect twelfth above (I love this effect). Then, the music makes a turn and effects a modulation to the relative. So, this section is only one measure shorter - the extra measure of stretto overlap - because the middle entry was shorter, but the, "interlude" was as long as the previous episode.

In the relative major we get another measure of overlap, and it sounds very sunny and optimistic, which is a nice respite from too much... seriousness. ;^)

The final episode is like the first two, but starting out from the relative. It is also truncated an additional measure, so this section is only eleven measures. Finally, it melds nicely into a miniature pedal point, which is quite effective.

At 55 we're back to the tonic minor, so this is the recap, and I used the closest two-part canon for it. I believe the word is splendifferous.

At the end of the canon, I grafted the final measure of the countersubject onto the subject to get a six measures canon. That false ending is, well, false. That's why it's not emphatic.

The grand finale is a two-voice hyper-stretto - both parts start simultaneously - between the subject above, and the augmented subject in the bass. This was originally a four-voice texture, so that makes this two voice extraction very, um, dramatic. The pitch climax of the fugue is the F at the beginning of 64. The interval over the bass is a minor ninth plus an octave. This is super-pungent, but it works because the listener knows the harmony is a V(m7m9) sonority.

At 66, the interval is a diminished eleventh, plus an octave. This obviously implies an augmented triad, so this came out amazingly well. After the subject finishes, it launches into a rising sequential line that becomes chromatic in the second half of 69. This leads to a rousing final cadence confirmation on the bottom system. Discovering that written-out trill variation at the end was a nice moment.

So, eighteen years after I wrote the string quartet fugue, I figured out how to reduce it to have the least possible number of elements as a two-part invention, and to get it to work for the guitar.

*****

Now, I could turn this into a two-voice fugue with the answer at the twelfth just by using the tonal answer a fifth higher in measure six (And then composing a counter-answer, of course). That would work, but it would introduce an imperfection because you would only hear the answer in the original exposition: A new section will be necessary, which you'll see in the next post.

Sibelius 7 Sucks

2012-01-07T19:38:00.003-06:00

Well, I talked myself into buying Sibelius 7 because the classical guitar sound they have is marvelous (I heard another composer's score with that sound). Problem is, THERE IS NO SOUND OUTPUT. I really, really hate stuff like this, because it's 2012 and there is no excuse for user-hostile interfaces anymore. Especially on a program that I spent several hundred dollars on.

My test for notation programs is simple: Can I figure out the basics intuitively, without needing manuals or tutorials? If not, it's a failure. Sibelius is a failure, just like Finale was (Another program I spent several hundred dollars on only to toss in the trash after a few hours of miserable frustration). Only Encore and Guitar Pro 5 have ever passed my test, so those are what I use (GP5 only when I need TAB, which is hardly ever). After using Encore for nearly twenty years, it is still the only one that allows a fast enough workflow for a COMPOSER/ARRANGER to use. Sibelius and Finale are only for digital typesetters, publishers, and those into self-flagellation.

I figured this might happen though, so I was prepared to only use Sibelius for the sounds by importing Music XML files and MIDI files I created with Encore. Even that failed.

When I launch Sibelius, I get that wretched, blasted Sibelius schmaltz, so I know the output is working, but when I press play, no sound. I went through all the steps on the AVID Help page for Sibelius, and still nothing. Poppycock.

Now, if I chose MIDI instead of Sibelius 7 Sounds I can hear a piano sound way back in the barely-audible range, so I know the MIDI file is playing back, JUST NOT THE SINGLE SIBELIUS CLASSICAL GUITAR SOUND I SPENT $500.00 FOR!

When will I ever learn: Encore is the only notation program that a composer can use.

UPDATE: It's worse than I thought. After spending a couple of hours reinstalling the program from scratch (There are FOUR discs), I did finally get sound... but only a piano sound, far, far away. Guess what? There's no way to permanently change the sound without changing to a specific guitar staff. You can change the sound in the mixer, but it will sound an octave high because... SIBELIUS DOESN'T PLAY 8va/8vb CLEFS! Can you believe that BS? Since all I wanted Sibelius for was formatting and playback, this is a deal breaker and I'm going to ask for my money back. As you regular readers know, I notate complex guitar music on a grand staff with 8vb treble clefs, like so:

Well, in the Sibelius System, in the Evil Trek Universe, where Spock wears a beard... this will always be piano music.

I can't believe AVID sacrificed the sequencer capabilities in what is supposed to be a world class product FOR COMPOSERS. Everybody who uses Finale and Sibelius scoff at Encore because it's so, "primitive." Well, AT LEAST ENCORE PLAYS BACK TRANSPOSING CLEFS WHEN YOU USE THEM.

I hate to curse on MMM, but Sibelius is bullshit, and I'm going to ask for my money back.

Happy New Year from MMM

2011-12-31T14:27:00.000-06:00

Here's to a great 2012 for everybody. Unfortunately, I'm still dealing with my ill mother, and though she made it through the last scare, she is in hospice care now as there is nothing left to do for her. We did have a nice Christmas eve together though, so that was nice, but then she crashed again on Christmas day. Ugh.

Looking forward to happier times.

Blog Note

2011-11-30T22:27:00.000-06:00

I am accumulating a list of posts to do, but my mother is in the hospital and is not expected to survive. So, my schedule has consisted of nothing but daily practice, working out, and then sitting with her. Needless to say, my heart isn't in posting right now.

Guest Post at Deft Digits

2011-10-13T00:27:00.001-05:00

Joe Walker, who I first discovered at the excellent From the Woodshed site, has a wonderful new blog called Deft Digits - GREAT guitar blog title: Wish I'd thought of that! - and he has a series going over there called, Discover a Guitarist. The idea is to expose his readers to guitarists discovered by we who have been around for a while.

Well, Joe paid me the distinct honor of asking me to contribute an entry to the series, and so I did. Since most of Joe's readers are rock and jazz oriented students, I attempted to broaden their horizons by telling the story of my discovery of Kazuhito Yamashita, who remains the most awesome nylon string player in the world, as far as I'm concerned.

So, head on over an poke around for a while - the previous entries are excellent too - and thank you, Joe!

No pretty girl for this post, as I'm bummed about the death of a long time non-musician inspiration, Steve Jobs.

R.I.P. Steve, and thanks for making my work easier and more fun.

Book Review: J.S. Bach's Musical Offering by Hans T. David

2011-09-27T19:27:00.000-05:00

I've gotten myself into a nice reading habit again - since I moved into my new house - so I guess I'm going through my music theory library yet another time.

This post is about Hans Theodore David's seminal book, J.S. Bach's Musical Offering of 1945.

David was the first, I believe, to figure out that Bach had an organizational scheme in mind with The Musical Offering: Previous to his work, most musicologists thought that the collection was a casual, even sort of random, collection of fugues and canons with a sonata for transverse flute thrown in. Dr. David presents, quite convincingly, evidence that there is a specific order to the pieces, and that this order reveals a very sublime and balanced plan.

What really sets this book above most works concerned with large works or collections, however, is the analysis. David's style is quite vivid and easy to follow - you'll want a study score like my little Eulenberg edition - and not much escapes his gaze.

So, if you want some serious insights into BWV 1079, look no further. This is for advanced students, obviously, but with that caveat, I can't think of any better way to increase your enjoyment and understanding of The Musical Offering.

Note to Garageband Users: Export Song to Disk ADDS DISTORTION

2011-09-06T12:27:00.000-05:00

I have been supremely stumped by a problem with Garageband: The projects sound clean when played back within Garageband, but the AAC/m4a files created under Share/Export Song to Disk sound absolutely horrible due to added distortion. I'm not talking about the usual audio quality loss due to compression from the project to AAC/m4a, but a huge amount of distortion which is especially noticeable in the lower registers.

Well, I have found a workaround: Instead of Share/Export Song to Disk, which creates the AAC/m4a file directly from the Garageband project, use Share/Send Song to iTunes. The song will then arrive in iTunes as an uncompressed AIFF file, which sounds just as perfect as the playback from within the Garageband project. At that point - in iTunes - go to Advanced/Create AAC Version and the song will be duplicated from the AIFF version to an AAC/m4a file. No more distortion! You can then drag and drop the AAC file from iTunes to your desktop, which effectively duplicates it, as the original stays in iTunes. Yeah, it's quite a few steps, but the AAC conversions Garageband does are so bad as to be unusable.

For some reason, Garageband does not convert from a project directly to AAC/m4a worth a darn, but iTunes does just fine converting from AIFF to AAC. No, I have no idea why, and yes, it is very un-Apple like.

Since this has been driving me crazy FOR SEVERAL MONTHS, I thought I'd put my solution on the intertubes for anyone else who is having this problem. I had actually gone back to Protools LE and even set up Cubase LE because of this problem, so it's nice to be back on Easy Street with Garageband: If you just record solo stereo guitar, you don't need the complexity of Protools or Cubase, not to mention the hardware limitations of Protools.

Now I can get back to my recording project!

Book Review: Beethoven's Ninth Symphony by Heinrich Schenker

2011-08-21T23:27:00.000-05:00

I bought this translation when it first came out back in 1992, so I've been through it before. After all these years, though, I had only vague recollections of it - almost all I remembered was being disappointed that there were not more insights into Beethoven's compositional practice - so I decided to give it another shot.

What I appreciate now that went right over my head before is that Schenker was responding to several egregious liberties taken with the Ninth by musicians who did not understand it, most notably the execrable Richard Wagner.

More than just a simple failure to understand the text of the score, what Wagner and those in his camp believed was that Beethoven's deafness made his orchestrations faulty. Schenker provided history with a much needed remedy to these wrong-headed ideas that goes beyond being merely a palliative: He actually cures the disease of thinking Beethoven's inner-ear was not up to the task of orchestrating in a very powerful and dispositive way.

About that powerful style: I also remembered being turned off by Schenker's mode of writing here, which is harshly polemical. Evidently, this is, if anything, more so with the original German - You could fill an ocean with the exclamation points in this book.

Now I understand why Schenker was so insensed [Dictionary says that's not a word - Ed.] (Yeah, but I use it all the time anyway - Huc.): He was trying to rescue this transcendent masterpiece from the ignorant hoi polloi who were desecrating it. For that, then, we owe Schenker a huge debt of gratitude. We take for granted now that these issues have been settled, but it was not always so. As late as 1912, when Schenker wrote the original of this book, many misconceptions about the Ninth still persisted.

If you want to get any "Shenkerian Analysis" insights from this, you can forget it: 1912 was before Schenker came up with those - IMO dubious at best - theories. Also being from 1912, the harmonic analysis is quite primitive, but useful enough as a guide that I intend to build on it next time through. Yes, I feel the need to go through this again, but next time with a full score (The only score of the Ninth I have is the Liszt piano transcription, which is great for analyzing the harmony, but impossible for the orchestration, obviously).

Not for the faint of heart then, but highly valuable from a historical perspective at least, and there are some cool compositional insights, they are just few and far between.

Composing Sonatas for Solo Classic Guitar: Part I

2011-07-17T18:27:00.001-05:00

The title of this post is a search term I see sometimes, and since I have figured out how to do this - and I'm currently learning a sonata I wrote - I thought I'd weigh in.

The first thing you need to understand is that I'm not referring to a complete multi-movement sonata, but a single movement composed with the sonata process. Secondly, you need to key on that last term and understand that sonata is a process - like fugue - and not a form per se. I tend to get a bit agitated when folks employ terms like, "sonata form" or, even worse, "sonata-allegro form." A sonata is not a form, or like a mold that you can pour music into, but rather it is a musical work that results from a process in which the musical ideas used determine the final construction of the piece. Sonata is also not limited by any Italian tempo designation!

If you reduce the sonata process to a single word, that word would be contrast: Contrast of any, several, or all musical elements; theme, motif, key, mode, meter, tempo, dynamics, register, texture, instrumentation, orchestration - whatever you want or that is appropriate to your goals. The possibilities actually are endless and limited only by your imagination, which is why the sonata process is the ultimate musical challenge, even beyond fugue (And, of course, you can combine the sonata and fugal processes!). Sonata can and should be overwhelming to those just starting out on the compositional journey, and the serious student is going to approach the matter with all solemnity: It's deep, man. lol.

For the composer-guitarist, the challenge is even greater than for most: Guitar as an idiom has many limitations that, for example, the keyboard idioms do not. This is why there is no such thing as a good guitar sonata that was composed by a non-guitarist without at least having a guitarist as a co-writer or consultant: Learning the idiom alone takes years.

So, if you are a guitarist reading this post who wants to learn how to compose sonata process movements, you should have several years of playing under your belt so that your understanding of the idiom is intuitive. You should have also composed many miniatures before you take this step into expanded formal proportions. Probably needless to say, knowledge of harmony is required, and solid counterpoint too: To merely be ready for this takes several years of dedicated work.

If you pass these requirements, what is the first step towards the sonata process? The answer to this question is the sonatina, which can actually be treated as a form with the following elements: 1] An exposition containing two different musical themes in two contrasting keys, 2] A repeat of that exposition, 3] A brief bridge, and finally, 4] A recapitulation in which the two themes are restated with the second one now in the original key. An introduction and/or a coda is optional. If the sonatina is in a major key, the contrasting theme will be in the dominant, and if it's in a minor key, theme two will be in the relative major. That may sound relatively simple, but for the guitar it is more than enough, because on the guitar it is often physically impossible to play the two contrasting themes in the same key at the recapitulation! That should give you an idea of what a monumental compositional challenge a sonata process piece for the guitar actually is.

Now, instead of starting out with the first sonatina I wrote back in the early 90's, I'm going to go through the first sonata I completed - which was not until 2005! - and then present the two sonatinas that lead up to it. That way, you'll get the idea how complex the issue is, and then how to logically approach it.

*****

I've gone through this piece previously on MMM, back in 2008, and I did a pretty good analysis in that post, so I'm not going to sweat the theoretical details here, just the problems I encountered and the solutions I came up with for them. Since I have now put the fingerings and position indicators in the score, any guitarist can now read through it to see, hear, and feel the idiomatic nature of the music.

Here is the MP3 for the piece, which is a MIDI to MP3 conversion I did in iTunes using a classical guitar soundfont. Later, of course, I'd like to add a recording, but that will have to wait. I plan to use real recordings of the sonatinas, though.

The piece begins with a sixteen measure introduction, which I wrote after I had gotten the exposition and counter-exposition done. Note that I originally called this movement a sonata, but it is now described as a fantasia: I figured the latter term was both less burdensome and more accurate: It's still a sonata process piece, but it has several, "fantastic" elements to it (Which owe their existence to the idiomatic requirements of the guitar).

As I said, I did a fairly detailed analysis of this previously, so I'm just going to mention the most salient points: The first eight measures seem to go toward a resolution to C major, but this is thwarted by the second half, which walks everything back to a preparation for A minor: These are the only two regions in the piece - C and A - and I use both the major and minor modes of each. The first eight measures are also in 3/4 time, while the second eight are in 2/4: These time signatures are also elements of contrast that I use. An introduction should set up the coming drama, which this one does perfectly.

The first theme area in the exposition is in A minor and 3/4 time, but there are some curiosities present.

I should mention here that there are two broadly different approaches to composing sonata themes: The more modern motivic method, and the older - and simpler - melody-and-accompaniment style. This is obviously a melody and accompaniment first theme, and I chose this exactly because it was simpler - this is my first actual sonata process piece, remember - and also much more idiomatic for the guitar: With the exception of the lead in to the dramatic pause in measure 24, this is all quite easy to play. Go ahead and try it!

The aforementioned dramatic pause is accompanied by an added measure of 2/4, which intentionally disorients the listener, but also adds an element of organic plasticity to the phrasing, which is a major aspect of this piece. At the melodic peak of this section in 27, the piece returns to 3/4 but very quickly moves to introduce the second theme, which requires more organic metric modulations at the end there in 30 and 31. As you can see, there at the end of the page, the contrasting theme will be in 2/4 time and the key of C major.

After the homophonic first theme, I chose as a contrast a more contrapuntal texture, but it is counterpoint at the relatively relaxed ratio of 4:1, and I kept it completely idiomatic to the guitar by having the faster melody always on top, and the third voice, such as it is, is simple zero-axis repeated-note accompaniment. More phrasing plasticity is found here, however, with the harmonies in measures 36 and 37, and then after the dissolve from 42 into the 5/4 measure of 43 which sets up the repeat of the exposition. Note that it appears to be a setup for a modulation to C minor until the last quarter note: C minor will appear later. Foreshadowing future events is a powerful emotional and structural tool in the sonata process. This page is also quite idiomatic and simple to execute, so pick up a guitar and play through it.

Because the sonata process evolved from the sonatina form - in a nutshell: it was actually pretty convoluted - early sonata process pieces had unvaried repeats of the exposition. That's what you find in Haydn and Mozart for the most part, but by later on in Beethoven's career, and through Brahms et al, there are actual counter-expositions present that contain varied elements. Obviously, I chose the more fully developed sonata process technique and this is an actual counter-exposition.

The first theme is now presented in the parallel mode of A major, but after the dramatic pause the second theme enters in A major, a sixth above the original appearance in C major. This is very unusual, but it's not a resolution of the second theme to the home key, as that would have to be A minor: This is a setup foreshadowing the recapitulation. Even this higher version of the second theme is not overly difficult to play, so give it a shot. I worked very hard to make this piece idiomatic and fairly easy to play.

Here we start with the second phrase of the second theme in the higher register, and that requires a longer dissolve and re-transition to the second phrase of the first theme, which starts in measure 70. Even though it starts out in A major, that second phrase ends up with a setup to C minor again, only this time we actually get C minor for the second theme back on its original level.

Putting the second theme in the minor mode makes it marginally more difficult to execute, but there are still no outstanding technical challenges, and since the guitar is so restrictive as to what themes will work in what keys - and even modes sometimes! - it is a good device that you can get a lot of mileage out of. It sure turned out to be effective in this case.

As you can see at the end of the page here, the piece seems to have capitulated to C minor, at least for the development, since the counter-exposition ends there with the double bar line.

Now for the development.

The concept of the development section was the toughest nut to crack for me: When the possibilities are endless, it's impossible to make a decision! The trick, then, is to figure out how to limit the choices. If you are just at the sonatina stage, a simple bridge consisting of a short harmonic progression is all that is required, but in a sonata no such evasion is possible. Also, since this sonata has two melodic themes, broadly speaking, and is not motivic in nature, actually combining the themes doesn't work.

What to do, what to do, what to doo-wacka, doo-wacka, doo?!

It took me years to answer that question: I had the exposition as far back as 1996, but I didn't come up with a solution to the development until 2005 - nine years! After finding the solution for the counter-exposition, I wrote the introduction, and that's when it hit me: If I wanted to exhibit contrasts in the development, the contrasts could be the textures instead of the themes.

In order to demonstrate contrasts of texture, I needed to make the development a theme-and-variations section, so I composed a six measure harmonic continuity that is closely related to the harmony of the first theme, as you can see from measures 89 through 94. There, I'm just presenting the new theme variant using the main texture of the first theme.

In the second variation, I use the sounding second texture that is so prevalent in the exposition and counter-exposition, and for the third variant, I use the sixteenth-note harmonic plucking pattern from the second theme area (And the introduction, I might add), and I put that back in A minor via the classic, "deceptive" movement from G(m7) directly to A minor: Limiting myself to just the levels of A and C made things much more manageable, and it also set up the architectural solutions, as you shall see. I also pulled off a slick metric modulation from 3/4 to 4/4 in measures 99 and 100, so I'm still contrasting duple and triple meters.

I only changed two notes in this piece when I went through and did the fingering, and one of them is the low open E in measure 106: That was an octave higher and WAY too difficult to execute previously (Actually virtually impossible on a non-cutaway classical guitar). This is much easier on a cutaway, but a non-cutaway... well, I don't play those anymore. lol.

Variation four, which starts at 107, is a double-time variant in 2/4 that uses the main texture of theme two. All of the open strings in the bass make it quite simple to play, actually, and I re-modulate back to C minor for the closing variation at 113, which is just a foreshortened version of the double-time variation just heard. So, the development is basically five variations of a harmonic continuity that contrast textures previously heard in the exposition and counter exposition.

To make it obvious that we are returning to the recapitulation, I used the second half of the introduction to lead back to the beginning... but I had some structural issues to deal with, which also took a lot of thinking-through.

So far I've presented the first theme in A minor and A major, I've presented the second theme in C major and C minor, and I've presented the second theme in A major (The higher register version of the counter-exposition). In addition to that, I've presented a third theme and variations of it in C minor and A minor in this development.

Well, I originally had the recapitulation begin with a return of the original theme in A minor, but I didn't want to hear it again: I've heard it twice, in A minor and A major, so I really didn't need to present it a third time. What I haven't presented is the first two variations of the third theme from the development that are in C minor in the home key of A minor: This worked out fabulously well.

So, instead of a third return of the first theme, we get a return of the third theme, now resolved to the home key. If the thematic elements are to be presented in twos - with the exception of the development variations that were already in the home key - then I still have theme two in the higher register in A minor to present, and that's just what I do. This has a truly awesome effect! At the point where the deceptive movement happened in the development, we get the lone forgotten thematic element! Truly, I love this. I think it's devastatingly effective if the listener has been paying attention.

After the second theme in the higher register ends, the re-transition figures return (Also for the second time), and there is just one duty left to perform: Present the first half of the introduction a second time, but this time as an ending coda! The result is a perfect piece of musical architecture with a bi-fold logic to it that is also organic and perfectly idiomatic to the guitar.

Now, since this is the second movement of four, I didn't want the ending to be too emphatic: A possible continuation should at least be allowed for. So, I didn't confirm the final cadence, and the final measure is exactly the same as the first measure of the exposition, but with the leap up to the higher A at the end.

Next post I'll show how I worked up to this piece through writing sonatina movements.

A Tale of Two Bachs... and Books

2011-06-14T21:27:00.000-05:00

I have just finished reading two wonderful and very different Books about Johann Sebastian Bach.

The first, Evening in the Palace of Reason [Hilariously, the Harper-Collins blurb says the meeting took place in 1757, long after Bach had died, instead of 1747] by James R. Gaines, and the second is the monumental Johann Sebastian Bach: The Learned Musician by Christoph Wolff, probably the world's foremost musicologist specializing in the life of Bach.

I said they are very different, but I might as well have said that they could hardly be more different. Evening in the Palace of Reason centers on the lives of Frederick the Great and J.S. Bach as they lead up to their meeting in 1747, while The Learned Musician is all about Bach. I mean, all that is known and that can be surmised about Bach. The former is an engrossing, fast paced read, while the latter is exceedingly detailed - sometimes even excruciatingly so - but deeply satisfying in a different way, especially if you are a serious student of the music of Bach, as I am, or a very curious and already well informed amateur.

Interestingly, the two authors have radically different takes on the 1747 Potsdam meeting as well as the resulting work, the famous Musical Offering. In Gaines' dramatic and humorous look at the situation, Frederick and Bach's son Carl Philipp Emanuel conspired together to embarrass the old man by coming up with The Royal Theme, which allows for virtually no stretti. According to him, Bach's ass was chapped by the obvious ruse, and The Musical Offering was Bach's revenge. Wolff, on the other hand, views Bach as a peace emissary (!) just months after Frederick's Prussian army had occupied his little burg, and The Musical Offering as a pious gift to the king.

Who is right? It's obviously impossible to be completely sure, but even the picture Wolff paints of Bach is of a larger than life character who did not suffer fools gladly, and who took perceived affronts to his music very, very seriously, so I'm inclined to side with Gaines. Besides, it makes for a much more interesting story.

So, I would recommend Gaines' book to anybody interested in Bach who likes fast and fun reading - his description of Frederick's "gay in all senses of the word" court is particularly humorous - but a word of caution is in order for Wolff: This is the deep end of the musicological pool here, and even I found parts of the book an epic slog (As someone who has analyzed in excruciating detail a lot of Bach's music, and who can actually, you know, compose in those styles, I found his descriptions of Bach's compositional approaches particularly eye-roll inducing).

Bottom line, though, I'll undoubtedly read Wolff's book again, while I think once through Gaines' tome is enough.

Six Year "Blogversary": New Demo Recordings

2011-05-14T21:27:00.001-05:00

Hard to fathom it's been six years this month since I started MMM, but it's true.

Had my first gig in nearly a year last Saturday, and it went great. It was a private party a bud of mine recommended me for because he couldn't do it. Amazingly, that was six months to the day since I picked the guitar back up and started rebuilding my set again after my mother's illness (during which I didn't pick up a guitar at all for about four months).

This also coincided with a transition in my practice regimen that has had me practicing for up to 6 hours some days, so it is time to start working on a new demo CD while I enter the next phase.

My MO for this is to just record three takes of each piece and save the best one of each, no matter what. In another few weeks, I'll do this again, and in a month or so a third time. This will give me three best-of-threes to chose a best of nine out of.

Without further ado, here are the first round versions.

Tears in the Rain - Joe Satriani

It's not as easy as it sounds.

Desert Song - Eric Johnson

This is about 85% of where I want it to be.

Spanish Fly - Eddie Van Halen

The first half sucks but the second half is about as good as I've ever played it.

A Day at the Beach - Joe Satriani

I'm pretty sure I'm the only nylon string player in the world who performs this. This is about 90% as good as I've ever done it, and I was actually amazed to capture this take. Recording this would be impossible without the RMC Polydrive because it filters out the percussion booms I create on the guitar (You have to beat the hell out of nylon strings to get them to ring, and I have large callouses on my right hand i and m fingers because of this piece). Flamenco guys hate not getting percussion through the pickups, but for recording - or performing - tap technique on a nylon string, a Polydrive is a requirement, not an option.

Eu So Quero Um Xodo - Dominguinhos

This is my favorite piece in my entire repertoire... that I didn't compose myself. lol. Not a bad take, but only about 75-80% of where I had it a year or so ago.

Stairway to Heaven - Jimmy Page

Yeah, yeah; but my demo CD just wouldn't be complete without it. The rock guitar solo section is rough, but not a bad single take, and I never put anything but single takes on CD's. Just weird that way.

*****

You know what? I've pretty much completely ditched ProTools LE for GarageBand. Since all I record is stereo solo guitar and solo guitar synth (Not on these takes, obviously), and since all my effects come from the MPX-G2 and FS1R, I don't need the complexity and hardware limitations of ProTools anymore. The real reason this happened is the Lexicon I-ONIX FW801s recording units I got: They sound a bazillion times better than any ProTools hardware I've ever had, and they are a 1U device. If you want the most extravagantly cool interface for GarageBand, the FW810s is it. I just used a single stereo instrument track and hit "delete" after renaming and saving each take. It's an incredibly fast and easy way to work if you record yourself.

Backup Gear: Two of Everything... at Least

2011-04-30T11:27:00.000-05:00

HAVING ONLY ONE OF SOMETHING LIKE THIS POLYDRIVE INTERFACE IS A RECIPE FOR DISASTER.

If you end up playing with either bleeding-edge technology or vintage gear - I have some of each - you figure out pretty fast that to be a confident and dependable performer, you need backup gear. For most devices that are not overly prone to failure, that means two, but for some critical things, it may be three, and for loudspeakers, it could just be one extra unit.

For example, when I leave for a road trip, I must have at least two guitars, two rack systems, and three PA speakers (I usually take four). In town, I usually just take an extra guitar if it's a background music gig, but if it's something major - for which I'm being paid a significant amount of money - I'll take an extra rack and speaker too.

This policy of having backup gear along has saved my bacon - and made me look like a hero - at several gigs. When that eight-year-old data backup battery dies and all of your programs vanish, there's nothing like being able to laugh it off and grab the backup rack out of your car or truck. Or, when that darned D-string breaks, it's sure a lot quicker and easier to just grab a backup guitar rather than changing the string. Of course, I learned this, like I learn everything in life, the hard way: I once travelled over 100 miles one-way for a gig and arrived with no 1/4" to Speakon adapters. Since I put speaker cables in each rack when I pack them up, a backup would have saved me that gig, the money... and just a ton of embarrassment: It's hard to forget to pack something twice.

That was my, "Never again" moment.

So, make a list, check it twice, and all your gigs will be very, very nice. Naughty waitresses are cool, but naughty amps are not.

The catch for me is - there's always a catch - I have two guitars that look exactly the same but that are in fact entirely different.

The Rider Nylon on the left has a high classical action, and the one on the right has a low flamenco action. I have decided that the classical action Rider is not a suitable backup guitar, because it's way to hard to play tap technique on it... so I need a third, of course.

But there's more to it than that. I love practicing tap tech on the classical Rider because it makes the pieces so much easier to play on the flamenco Rider, and I play gigs without playing Spanish Fly and A Day at the Beach all the time. So, I'm glad I got the classical setup, and I wouldn't want to be without it, but two is not enough in any case. Why?

Because, the real problem is with the RMC Acoustic Gold saddles, or perhaps just my less than stellar luck with them. Back when I was playing Godin guitars with these saddles, I had one go entirely dead on me - the G string on that guitar - which is where I got the idea to always have two guitars along. Well, these pickup elements are by far the best in the world, but to be so sensitive and clear, they evidently also have to be quite fragile. Lummox that I am, I've managed to break two of them while changing strings! Fortunately, both on the classical backup ax, and I managed to fix the second one myself.

So, you see where this is going: I imagine the scenario where I'm changing strings in anticipation of a road trip and I break one of the RMC saddles. I don't have time to get it repaired before I leave, so I have to hit the road with just one guitar. This is not acceptable, so I'm going to order a third Rider Nylon/RMC - my second in the flamenco setup - before the end of the year. I'm also going to order some extra saddles so I can repair them here myself, rather than having to ship them off to get the work done. Yes, having extras of things like that often means you'll never, ever need another one again. lol.

The good news is that I'm getting all of these ducks in a row because I'm again free to get on the road and do some serious performing now that my mom is being looked after. I've worked up to some six and seven hour practice days over the past 5 months, which will be the subject of some upcoming posts, and some problematic pieces I've been working on for years have suddenly come into focus. It's a great feeling.

She bears an almost uncanny resemblance to my high school sweetheart.

Epic Drunk Driving Fail in the 'Hood

2011-03-26T13:27:00.000-05:00

Some young buck in a pickup truck - no word on whether he was wearing a pink carnation, but he was definitively out of luck - launched himself and his vehicle into a house in my neighborhood back at the beginning of the month. The truck burst into flames and caught the house on fire.

I walk by this house every day on my 6-8 mile walks. Fortunately, nobody was home and even the drunk driver was not seriously hurt.

To give you an idea of how spectacular this was, here is the intersection from which the truck went airborne.

The cross street is Thousand Oaks, and the road going off into the distance is Bulverde Road, which turns into Scarsdale as it enters my neighborhood. What you can't see from this angle is that there is a rise, very similar in size and angle to a waterski jump, onto Scarsdale. The reason there is no damage to the landscaping in the foreground is because the truck was in the air over it. The police estimated the distance the truck was airborne at over 100 feet.

I walked across to the sidewalk in the picture above to show where the truck landed.

He made it almost to the sidewalk, and the curb deflected his trajectory to the right, and into the house. Within a couple of days, the house was demolished down to the slab.

This is the second time, to my knowledge, that this house has been hit by a car. I don't think I'd want to live there. lol.

The caption said she's French. Having spent a lot of time in France, I could have told you that just by looking. She's almost stereotypically Gallic. Quite pretty, though.

Two-a-Days

2011-02-27T21:27:00.000-06:00

No, not football training camp, technical and performance practice in two separate sessions every day. Grueling, but I'm about 75% ready to gig again... and, I've worked out twice a day every day this year so far too: Bowflex and a 90 minute walk.

Feeling much much better than I did last November.

Something for all the True Blood fans out there.

Pretty much the ideal redhead. Yeah, I watch the show. lol.

Hiatus

2011-01-29T06:27:00.002-06:00

Taking a little time off from the blog to catch up on some set development chores. My mom got very sick late last year and I wasn't able to play for a few months, so I'm rebuilding my set again. Yes, I rebuilt it at the beginning of last year too, so this is quite frustrating, but now that mom is taken care of, I ought to be back performing again by March or April. In fact, I have a standing two gig per week offer at an eatery already, so I'm anxious to get the set back together.

Life sometimes intrudes on one's plans.

MMM Merry Christmas

2010-12-25T07:27:00.000-06:00

Merry Christmas, readers. Hope you're looking forward to great things coming together in the next year, as I am.

A Lexicon Christmas

2010-12-17T15:27:00.000-06:00

Thanks again to Keith at The JukeYard for servicing my Lexicon MPX-G2's. I now have four out of five done up with fresh batteries and thorough checkups.

While doing this project, I lamented to Keith that it was too bad the MPX-G2's tuner wasn't able to calibrate to the A= 432Hz philosophical pitch I tune to. Well, it does! Sure enough, I had missed a level in the Tuner Edit Menu that allows offset in cents. It comes out that -33¢ is equal to A= 432 (For some reason, the Axon AX-100 Mk II zeros in on 432 at -34¢), so I was able to eliminate the Behringer BTR2000 Racktuners from my rigs, which has made them 1U smaller. Additionally, the Lexicon MPX-R1 MIDI Remote Controllers I use have a Tuner Function, so I'm able to use that again, which is far superior to having a separate mute switch for the Behringers.

Here's my day-to-day rig, now down to just 3U:

And for the larger venues, I have a Lexicon 212 - which is just a Bryston 3B-ST with a Lexicon faceplate - instead of the Bryston 2B-LP, which makes 5U (The SKB Rolling Rack was the only one of 5U I could find).

Finally, without the BTR2000, my synth rig has gone from 8U with a vent panel to 6U!

You can see why I called this post A Lexicon Christmas: Three MPX-G2's, an NT 212, and an I-ONIX fw810s.

Not in this post are the 4U rack, which uses a MESA Stereo Simul-Class 2: Ninety tube power amp, and my main recording rig, which is still a work in progress. Keith says I'm a musical mad scientist, to which I reply, MuaHahahaha!

Merry Christmas!

The Juke Yard: Lexicon MPX-G2 Service and Repair

2010-11-17T21:27:00.000-06:00

First of all, if you own a Lexicon MPX-G2, you need to join the STEC Records MPX-G2 Forum. The site administrator there, Bob Sellon, is a former Lexicon employee who helped design the MPX-G2, and that is where I met Keith of The Juke Yard, which is where Bob sends his G2's for service.

If you have a Lexicon MPX-G2, you ought to know that it uses a backup battery to keep the user programs in memory. When that battery fails, "poof," all of your programs are gone. Since I have played through nothing but MPX-G2's for over ten years now - it is the best combination preamp and effects unit ever made for guitarists still, after being discontinued for several years - I'm very paranoid about the backup batteries, because I have literally hundreds of man hours worth of programming stored in them. I say them, because I have five MPX-G2's now. One reason for so many is my paranoia about losing all of that programming work... the other is that I'm a gearoholic who, "needs" five different rigs for every conceivable type of venue and recording situation.

Well, that memory backup battery isn't just a regular watch type battery that you can easily replace, it's a special type that needs to be un-soldered to remove, and re-soldered to replace. I'm terrible with that stuff, so I had Keith do two of my units last week (I'll send two others to him after Thanksgiving). Turnaround was super-fast, the work was excellent, and he also checked out every function to make sure they were 100%. IMO, there's no substitute for the peace of mind I get from having a pro do this kind of work, so by New Years, My G2's will be set for another decade of gigging and recording.

Keith also works on other Lexicon gear - and juke boxes, of course - so if you have a dead Jam Man, send him an email and bring that puppy back on line.

But wait, there's more. STEC Records also has a dump tool for Windows users that allows for all of the MPX-G2 user programs to be dumped to a PC for safekeeping! I'm going to put Windows on one of my Intel Macs so I can use that and also the Windows-only FSeqEdit program for my Yamaha FS1R's.

So, I'm happy to welcome The Juke Yard to the, "Hucbald Endorses" section of my sidebar.

Learning to Play MIDI Guitar

2010-11-13T02:27:00.000-06:00

I've been controlling synthesizers with guitars since before MIDI was even invented. I saw this ad for the Synclavier guitar system in Guitar Player magazine and had to have one. The smallest 8-voice monophonic guitar system was $19.500.00... yeah, I had to scrape, scrounge, and sell a lot of stuff to even get in the door. That system eventually grew into a 32-voice stereo system with 8 MIDI outs, SMPTE, the Velocity/Pressure keyboard and a Terminal Support Option. It was ridiculously expensive, but getting it was the single best decision I ever made in my life. No way I'd be anywhere near the musician I am today if I hadn't.

Speaking of today, now it's easier than ever to set up a MIDI guitar system. Back in my Synclavier guitar days, we were working with the original Roland GR series guitars, and setting them up just right so that they tracked properly could be a nightmare. I went through at least three Roland GR guitars before I found the Steinberger GL2T-GR that ended up being, 1] an awesome playing ax and, 2] something that had the required Roland pickup.

Here's a knot-head with a GL2T-GR who has no earthly idea what he has or how to use it:

Hey buddy, I'll take that off your hands and pay you well. Sheesh, I'd be rescuing that poor ax. lol.

Anyway, that was back when you could only control a synthesizer with a steel string guitar, because all of the available hexaphonic pickups were magnetic. Now, with piezoelectric saddle designs like the RMC Polydrive II's that are installed in my Blackbird Rider Nylon/RMC guitars, we classical-based players can get into the act too (Not that any of the uptight, painfully self-conscious traditional classical guitarists would ever go out on such a limb).

That brings me to Point 1: The most important decision you'll make when you decide to play MIDI guitar is which guitar to use and most importantly, which pickup system to put in it. Even if you play steel string guitars, you'll be better off with a hexaphonic piezoelectric design in which the strings make physical contact with the transducers: There's nothing better - or more reliable - than a physical connection between the string and the pickup elements. No electromagnetic hex pickup will ever be as good.

One of the main reasons why this is the case is that the RMC systems isolate the transducers from the body of the guitars: If you are using them as just regular guitar pickups, you can hit the body of the guitar and no sound will come out of the amp at all. Flamenco guys hate that, of course, but without all of those sympathetic resonances coming from the body of the guitar, the interface has an easier time picking up the fundamental, and less chance of getting confused and mis-tracking.

So, settle for nothing less than an RMC Polydrive.

Now for Point 2: Which guitar-to-MIDI interface should you get? Before getting back into MIDI guitar, I researched every guitar-to-MIDI system that had been made since 1990. There are several usable units out there, but almost none of the older ones are optimized for piezoelectric transducers. The Blue Chip Axon AX 100 - which later became the Terratec Producer AX 100 Mk II - and the Terratec Axon AX 50 USB are really the only choices now.

I said now, but unfortunately Terratec has discontinued the Axon lines, and so they are an eBay-only item now. This has caused a huge spike in their prices - I saw an AX 50 USB go for $1,009.00 a week or so ago - so if you're smart, you'll try to land an older Blue Chip model. That's what I did: I got a Blue Chip Axon AX 100 SB (With the then-optional GM sound card) for half of what that AX 50 USB went for. Terratec Axon AX 100 Mk II's are going for around $1,500.00 right now: I paid less than that for NIB unit a year ago! So yes, I have two Axon AX 100's: A used Blue Chip and a new Terratec (One of the very last ones sold new, evidently).

So, you've chosen your guitar, gotten your RMC Polydrive installed, and snagged an Axon: Now what? If you got an AX 100 SB or an AX 100 Mk II, stop right there! Since they have onboard sounds - even if they are cheesy GM samples - you can keep your first rig dead solid simple and learn to play it. That's what I did. I just learned my way around the Axon for a year before I decided to add a more interesting sounding MIDI synthesizer sound module.

If you've read this far, I can give you the bad news now: Statistically speaking, I have almost no chance of being wrong when I tell you that, you will fail. It'll be too difficult to learn the Axon (You'll have to spend a lot of time reading the manual), you'll be unwilling to improve your technique so the synth tracks (You have to play very clean and deliberately), and you'll give up. Think about it: How many guitarists do you see out gigging with a MIDI guitar rig? My admittedly anecdotal experience is exactly none. Back in my Synclavier days, there was John McLaughlin, Pat Metheny, Al Dimeola, Frank Zappa... and me. And out of all of us, I was the only guy who played the Synclavier live on every song in the set.

The point is, you have to have a powerful need to succeed, you have to be willing to invest a lot of time learning, and most of all, you have to be resolved to radically altering your technique to make it all work. Almost no guitarists are ambitious and disciplined enough to do it, and that's just the way it is.

Now back to you Axon AX 50 USB guys. I really want one of those for the USB connection to my computer, but I'm not willing to spend over $1K on one. If you do have an AX 50 USB, though, there is no sound card in it, so you'll need something to play with it. In the keeping it as simple as possible department, any GM sound module like a Roland SC-88 would work, or you could use the USB connection to play the sounds inside your computer. Every Mac, for example, has a GM sound set in it.

My personal philosophy is that samples suck, and pure synthesized sounds are much more interesting. Ten plus years of Synclavier programming will do that to you. So, just as soon as I was comfortable with the Axon, I added a Yamaha FS1R FM/Formant Sequence synthesizer sound module to my rig - it's going to take a couple of years to become a virtuoso programmer of this thing - which is the best synth ever, if you like digital.

Check this out:

The formants allow you to program this synth to talk! Like I say, screw samples.

If you're into the old fat analog sound, there are just a gazillion older Roland analog synth sound modules you could get into, but remember to keep is as simple as possible at first and master one task at a time. No matter how good you are with this stuff, you can overwhelm yourself with a complex MIDI guitar system if you don't take it a step at a time. I've been playing synth guitar for twenty-five years, and when I got the Axon, I started over from scratch: Playing scales cleanly so the unit could track them.

If you end up one of the few winners in this game, you'll end up with a rig that looks ominously cool on a darkened stage before the lights come up (Hey, these considerations are important)...

... you'll be playing some very high tech guitars...

... and you'll be the hero who tamed the digital beast.

Good luck... you're gonna need it. lol.

Rack Building: The Mobile Yamaha FS1R Guitar Rack

2010-11-03T06:27:00.000-05:00

Sorry for the lack of posting recently, but since getting to the critical mass point in understanding and programming the Yamaha FS1R, I have been hit with an urgency to get a rig together so I can gig with it. This has involved a lot of buying and selling on eBay, which is a slow process. For one thing, obscure discontinued gear like the Yamaha FS1R and EDIROL UM-880 don't appear every day, and for another buying and selling in auctions just takes time. The savings are worth it, however.

The third FS1R still hasn't appeared, and neither has a UM-880, but I did find a UM-550 which will work for the home recording rig, and I had enough gear to get the gig rig to a working configuration. Here it is!

Top to Bottom:

1] 17" 7200 RPM High Res MacBook Pro
2] Bryston 2B-LP Power Amp
3] Behringer BTR2000 Racktuner
4] Lexicon MPX-G2 Guitar Effects Processor
5] Axon AX-100 Mk II Guitar to MIDI Interface
6] Blank Vent Panel
7] Lexicon I-ONIX FW810s Mixer/Recording Interface
8] Yamaha FS1R FM/Formant Sequence Digital Synthesizer
9] Furman AR-1215 Line Voltage Regulator

Dragging a 17" 7200 RPM High Res MacBook Pro to a gig might seem like a lot of trouble, but the Lexicon FW810s requires one just to mix - it unfortunately defaults to mono without a computer - and I've hated every other 1U mixer I've ever tried: The preamps are just horrible in the cheaper gear. Also, of course, I can record live using the MacBook Pro, which is a big part of the mission of this rig.

I detest every solid state power amp made except for Brystons, for reasons that ought to be obvious to the cognoscenti.

The Behringer Racktuner is still the only rack mount tuner that can calibrate to the A= 432 Hz that I tune to.

Even Roland can't touch the Axon AX-100 Mk II, but I understand that TerraTech may have discontinued it. Oh well. They wouldn't be the first company, by far, that has failed to make a guitar-to-synth interface financially viable: ARP was killed by the Avatar as far back as the 80's.

Eventually I'd like to have an EDIROL UM-880 MIDI interface/merger/patch bay in place of the blank vent panel so I can program this rack's FS1R too - and have this rig as an alternate recording environment - but right now it's a pure performance setup with MIDI thru from the MPX-G2 going to the AX-100 Mk II for program changes, MIDI out from the AX-100 Mk II going to the FS1R to drive it, and MIDI thru from the FS1R going to the FW810s for recording MIDI tracks.

I love the Lexicon FW810s. It has better preamps and a crisper, cleaner sound than any of the various Digidesign recording interfaces I've tried, and it's 1U: Digi can't seem to put a recording interface into anything less than 2U, and their preamps sound like crap, IMO. Overly bright and very harsh sounding.

The mighty Yamaha FS1R. The greatest digital synthesizer ever made. It can do a lot of things even my old Synclavier couldn't do, and it has higher bit resolution and a suite of onboard effects. There is no other synth that I think is worth my time to learn.

No cheap power conditioners for me. The Furman AR-1215 is an actual line voltage regulator with an isolation transformer. The kind of clean and steady voltage it delivers is even more important if you use analog synths or tube amps (Two of my other rigs use MESA/Boogie power amps).

The Lexicon transformer plugged into the face of the Furman powers a Lexicon MPX-R1 MIDI foot controller, which is not pictured. It allows me to control the rig remotely.

So there it is.

I'm currently working on the dedicated stay-at-home programming and recording rig, which will have two FS1R's in it. No telling when another pristine FS1R will appear on eBay, much less an EDIROL UM-880: Nobody who has a UM-880 seems willing to part with it.

Couldn't decide if I wanted to post a redhead or a brunette.

Yamaha FS1R Editor for Mac OS X

2010-10-19T09:27:00.000-05:00

A Musical Lifetime's Worth of Background:

It's been about five years since I sold my New England Digital made Synclavier Digital Music System, which I had for a full twenty years since 1984. I loved that thing because it was a digital additive/FM synthesizer that could make unique never before heard sounds, I could play it with a Roland GR pickup equipped guitar (I used a Steinberger GL2T-GR axe), plus it had a thirty-two track sequencer that operated just like an analog or digital tape deck, but with all of the editing advantages a virtual digital world brings.

For years after I got it I would spend hours every day pouring through the several phone-book-sized manuals and just messing with its synthesis capabilities. I got so good at it that within a couple of years, New England Digital was distributing many of my timber programs, including the sound effects that gave me minor fame, with every new Synclavier sold.

For an idea of where that curiosity about the nature of sound lead me, here's a piece I created with the Synclavier way back in 1994 - a full ten years after I got it - when I was a doctoral candidate at UNT. Almost every sound in that - with the exception of the church bell, I think - is something I created from scratch just starting with a sine wave and an FM ratio. Not only that, but all you'll hear in that is the Synclavier: No reverb, effects, or even EQ. That was recorded straight out of the Synclavier's stereo outs into a DAT deck.

Of course, I also have the Synclavier to thank for making me aware of the fact that music works because the overtone sonority is a dominant seventh chord. There's nothing like directly messing with the harmonic series through additive digital synthesis to make one aware of the true nature of sound: The Synclavier enabled me to mess with both the amplitude and the phase relationships of the harmonics out to P32, apply FM to the resulting waveforms, and then to even crossfade a series of them together (Which is how I created my groundbreaking sound sculptures like in the background of this Synclavier sequence).

Well, when I switched to nylon string guitars in 1988, I initially went through a back-to-basics pure acoustic phase as a player. I still used the Synclavier for electronic music composing, but I wasn't really interested in playing it with a guitar anymore. My acoustic phase was short lived, however, as I soon missed my phase, flange, chorus, and reverb effects. Plus, performing with an acoustic classical guitar is a loser's game, as even polite conversation can drown it out. So, by 1990 I was experimenting with electric nylon string guitars, and finally in 1998 Lexicon came out with the MPX-G2, so it took a full eight years just to find a preamp/effects unit I liked.

As for electric nylon string guitars - sheesh, what a grueling ordeal that has been - I started out with the only game in town at the time, which was a Gibson Chet Atkins CEC. It weighed a ton and was a pain to EQ. The Godin's were the breakthrough because of the RMC Polydrive hexaphonic pickups... which can run synthesizers. But it wasn't until I discovered the Blackbird Rider Nylon guitars just last year that I finally had an axe that I was over 90% satisfied with (The Rider's fingerboard is radiused, whereas it ought to be flat). The Rider Nylon can also be had with the RMC Polydrive, of course, and I now have two of them and play them exclusively.

Having the sound system and guitar sorted out - a process that took twenty years! - naturally lead me back to thinking about playing synthesizers with the Polydrive equipped Rider Nylons. So, I got a Terratec Axon AX-100 Mk II guitar to MIDI interface, but it just had a GM sound card, and wasn't a synth. that lead me to discover the Yamaha FS1R formant sequence/FM synthesizer, which I had never even heard of before last year. It had slipped under my radar between the time I stopped playing the Synclavier and when I started getting interested in synthesis as a performer again.

I was fortunate to find a virtually new one on eBay - they were only made for a couple of years - that had never seen a rack, and so I was off - or so I thought. Like a lot of Yamaha products - even the classic DX-7 - the programming architecture was tortuous to navigate. It was even worse than the DX-7 because everything was compressed onto a 1U rack faceplate! Additionally, there was no way to edit the formant sequences even on the unit itself: A major flaw, as the formant sequences were one of my primary points of interest.

Back in my Synclavier days - after the MIDI option came out for it - I had eight MIDI outputs to work with, so I paired it up with a Yamaha TX-816, which ammounted to eight DX-7's in a 4U rack chassis. Well the individual TF-1's - that's what the DX-7 modules were called - had no programming controls at all on them, so this was the first thing I ever encountered that required a computer software editor. Back then - we're in 1986 - the best one in terms of intuitive usability ran on the old Commodore 64: That was my first experience with a GUI, and it was earth-shattering (I wasn't a Mac guy then, obviously). It worked fantastically, and I learned the Yamaha carrier/modulator algorithm architecture, so I already knew a lot of what is in the FS1R - it can actually load play DX-7/TF-1 programs - but I needed an editor.

Well, a link at the bottom of the Wikipedia FS1R page lead me to the website of Japanese Mac programmer and synthesizer programming virtuoso K_Take.

K has written a fantastic FS1R Editor for Mac OS X, and it's available as a free download!

Minor Teething Pains

There were a few problems, as I don't think the FS1R Editor will work with FireWire MIDI interfaces. At least, it wouldn't boot with my Lexicon I-ONIX FW810s. So, I got one that was in a screenshot on his website, the EDIROL UM-2ex. After installing it and downloading the advanced driver, it worked perfectly. Now, however, I have to find one of the discontinued UM-880's for a rack mount version (I need one for my growing studio anyway).

I finally got it all set up last week.

The G5 PowerMac is running the FS1R Editor, the M-Audio KeyRig 49 - I program sysnths with a keyboard, not from the guitar - the UM-2ex, and the lower of the two FS1R's in the rack. That's right, I now have two FS1R's; one to program, and one to play with the guitar.

The MacBook Pro runs the FW810s mixer, and so it does the guitar rig part of the rack, which is everything else in it. I also record with the MacBook Pro using the 810s and Garageband or Cubase LE.

As you can see, it's a tight setup, and I can perform, record, and program without moving at all. The tiny Apple wireless bluetooth keyboard and the Kensington trackball were absolutely essential, as was a USB keyboard of not more than four octaves plus the obligatory semitone.

Eventually - when I start performing with this rack - the second FS1R will split off into a dedicated stay-at-home recording rack, but right now it's excellent for programming and having a second unit to back everything up to (I had to replace one battery already, which is thankfully very easy).

The FS1R Editor GUI

When you first launch the FS1R Editor, you are greeted with the MIDI Interface Setting window. The very first time you do this, you have to use the drop-downs to select everything. After that, it comes up just like this, unless you forget something like powering up your keyboard, in which case you have to quit, turn it on, and relaunch: It won't recognize a keyboard that is powered up at this stage of the game.

After you click OK, FS1R Editor scans the FS1R, which takes several seconds, and then greets you with the Performance page of the editor, which actually has four pages in it. This is the Part Parameter page, and since an FS1R Performance program can have up to four parts, you select those with the buttons at the top left of the recessed field.

The MMM template is 700 pixels wide, and the actual window is 1000p in width, so this is 300 pixels narrower than actual: It's plenty big enough to work on comfortably, but a trackball does help to accurately grab and move the small sliders.

The second Performance page has the Formant Sequence selection and control. yes, you can create formant sequences but I haven't tried that out yet.

Master tuning and transposition is on the right - very handy since I tune to A=432 - and you can also transpose the entire unit here.

One of the very best things about the FS1R is that it contains a full suite of onboard effects: Reverb, phase, flange, chorus and three band parametric EQ. K hasn't made the FS1R Editor anything like graphic-intensive, but the routing schematic is particularly useful here, as is the EQ shape and scale.

The final Control page of the Performance section allows for assignment of the knobs and MIDI controllers, and their sensitivity settings. There's a helpful Default Setting button for when you get into, "Oh my God, what have I done?" territory.

The Voice page is where most of the fun happens for me, as this is where you create sounds. It took me a minute to notice, but the Common Parameter box on the left actually has six pages to it, which are the sideways buttons along the left. This is unusual, but I'm sure K did that because they wouldn't all fit horizontally along the top. The LFO box is what you get when you first hit the page.

Believe it or not, the programming section on the right reminds me of a high resolution version of the old Commodore 64 program I had to program the TX-816: very simple, straight forward, intuitive, and self-explanatory. The graphics are just what's needed: The pic of the algorithm selected, the shape of the envelope, and the Frequency-based Envelope Generator.

Yes, we have filters, just like an analog synth.

And the filters have envelopes, of course, which is another nice place for a simple graphic representation and intuitive faders.

The Pitch Envelope Generator.

Here's the Formant Control section. It's a mix-and-match section for the internal formants unless you initialize one from scratch (As I understand it at this point, which isn't very well yet).

And finally, the FM Control box. You select the voiced and unvoiced operator to work with to the right, just under the algorithm graphic, by the way.

There's an on screen keyboard you can use if you don't have a USB or MIDI keyboard, but I found this very unsatisfactory to use. I need a physical keyboard for immediate feedback.

To summarize, this is a very compact and elegant little program that is an absolute must have if you have an FS1R and wish to program it. Can you imagine how long it would take to get to some of this stuff just using the FS1R's faceplate controls? Me neither.

Much more to come about this, of course.

Why Music Works: Chapter Fifteen

2010-10-18T17:27:00.000-05:00

The Integrated Chromatic Contextual Systems: Chi, Psi, and Omega

PREFACE to All Posts:

This is to be the culmination of the Musical Relativity series of posts I did back in 2006, which can be found to your right in the sidebar. Back then I was calling the series Musical Implications of the Harmonic Overtone Series. Even before that, I did a series of posts called Harmonic Implications of the Overtone Series that started this all. Here, I am presenting the final weblog version of the evolving book I've decided to publish with the intention of getting some feedback before I create the final print version, which I plan to put into the ePub format for iBooks. So, please feel free to ask any questions about anything that you think I haven't made perfectly clear, and don't hesitate to offer any constructive criticisms or suggestions. Since this project is the accidental result of several decades of curios inquiry - and many prominent and also relatively anonymous theorists and teachers have contributed ideas to it (Which I will credit where memory serves and honor dictates) - I am eager to get a final layer of polish from any and all who may happen to read this series and find it useful, or potentially so. Since I am creating these posts as .txt files first, revision should be a simple process.

Since my pre-degree studies at The Guitar Institute of the Southwest and my undergraduate work at Berklee College of Music looked at music theory from the jazz perspective, and then my master of music and doctor of musical arts studies at Texas State and The University of North Texas were from the traditional perspective, a large part of how I discovered the things in this book-in-progress was the result of my trying to reconcile those different theoretical viewpoints. Since I want this work to be of practical value, I have retained all of the traditional theoretical nomenclature possible, and only added to it where necessary to describe phenomena that have not heretofore been present in musical analysis. I have, however, standardized terminology into what I think is the most logical system yet devised, and that will be explained as the reader goes along. There is a lot of built-in review and repetition - something I've learned from my decades of private teaching - so even a once-through with this systematic approach to understanding musical phenomena ought to be of significant benefit.

Outright addition to traditional musical analysis is limited to the symbology required to label root motion and transformation types so that the root motion and transformation patterns are visible: This greatly facilitates comprehension, and since good and bad harmonic continuities are separated by the effectiveness or lack thereof in the root motion and transformation patterns, this also actually functions as an aid to composition. All symbology - old and new - has been worked out over the past three decades so that everything is readily available with the standard letters, numbers, and symbols found on a QWERTY keyboard.

Finally, for the contextual systems, I have used the Greek alphabet: The normal diatonic systems - those comprised of two minor seconds and five major seconds - are Alpha, Beta, and Gamma. The exotic diatonic systems - those that contain a single augmented second - are Delta, Epsilon, and Zeta, and finally, the alien diatonic systems - those that contain two augmented seconds - are Eta, Theta, and Iota. Since the theoretical writings that started western art music out were handed down from ancient Greece, I thought this would be a fitting tribute, as well as a handy and logical classification scheme.

Basically, if you have a baccalaureate-level understanding of music theory from either a jazz or traditional perspective, you should have no problem understanding anything in this straight-forward treatise.

INTRODUCTION to Chapter Fifteen:

In chapter one, I demonstrated how the overtone sonority generates the three normal diatonic systems - those seven note systems that contain two semitones and five tones: Alpha, Beta, and Gamma - and then in chapter two we examined each of those systems in detail, discovering that the primacy of Alpha is due to the fact that all seven of its harmonies can be arranged in progressive order. In chapter three, we examined the contextualization of Alpha Prime, looking at the various different root progressions types it can exhibit, their various transformations, and through this we also started to look into the world of musical effect and affect. Chapter four was dedicated to examining how Beta Prime and Gamma Prime compared to Alpha, using the same musical proof formats developed in chapter three. Through those proofs, we discovered some very unusual harmonic effects that evoke the uncanny that are contained in the Beta and Gamma systems. Chapter five then took us out of the diatonic harmonic world and into the chromatic realm as we discovered the origins of the secondary dominant sub-system sonorities. After the secondary dominants, in chapter six, we looked at the secondary subdominant sub-system of harmonies, which completed a larger set of integrated chromatic systems, which we will look at in detail later.

Then in chapter seven, we looked at the exotic diatonic systems - those seven note contextual systems that contain a single augmented second: Delta, Epsilon, and Zeta - and in chapter eight we looked in detail at the root motion types they contain, and the unique harmonic effects that these unusual systems create. With the exotic systems out of the way, in chapter nine, I was free to demonstrate a phenomenon that is an artifact of patterned root progressions, which I pointed out earlier, and that is harmonic canon. Depending upon how harmonic canons are developed and set up, I showed how they can also exhibit the phenomena I call Musical Escher Morphs and Harmonic Mobius Loops. Returning to diatonic contextual systems in chapter ten, I introduced the alien diatonic systems - which are those seven note systems that contain two augmented seconds: Eta, Theta, and Iota - and then its companion, chapter eleven, examined the isolated root motion and transformation types in those alien systems.

After finishing analysis of the nine diatonic contextual systems with the comparative morphology example in chapter twelve, we then turned to the extra-diatonic contextual systems of Kappa, Lambda, and Mu in chapter fourteen. The penultimate chapter in this section of WMW, chapter fourteen, examined just versus equal temperament, and so now in fifteen - which is the final chapter in this section on harmonic aspects of music - we will put the secondary dominant and secondary subdominant sub-systems together into the integrated chromatic contextual systems of Chi, Psi, and Omega. I skipped to the end of the Greek alphabet here because there may be some other systems to add in the future (But I got every one that can be generated by the overtone resolutional paradigm, so far as I know).

*****

CHAPTER FIFTEEN:

EXAMPLE 73:

Listen to Example 73

Starting on the tonic of Alpha Prime, the resolutional paradigm takes us immediately to the primary subdominant, and then into the secondary subdominant contextual sub-system at the next resolution to bVII(M7), which is an Alpha 4/Lydian sonority. Further Lydian sonorities follow as we traverse the secondary subdominants through bIII(M7), bVI(M7), bII(M7), and finally bV(M7), which is an enharmonic sharped fourth degree. The next enharmonic progressive resolution to V(m7)/iii puts us into the secondary dominant contextual sub-system, and we traverse V(m7)/vi, V(m7)/ii, and V(m7)/V before arriving at the primary dominant preparation for the final progressive resolution to the tonic. So, the secondary subdominant contextual sub-system acts as a huge subdominant preparation for the most remote of the secondary dominants, which in turn act as a gargantuan dominant preparation for the return to the tonic: For centuries this was reduced to the kernel of I, IV, V, I, with the super-progression from IV to V literally short circuiting what the overtone series implied was the ultimate outcome for a major locus integrated chromatic contextual system.

Note that the V(m7)/iii does not have a diminished fifth when we follow the resolutional paradigm through the secondary subdominant sub-system: The paradigm rules that the root of the targeting chord is retained as the fifth of the target chord, so the G-flat is sustained as the enharmonic equivalent of F-sharp into the B major/minor seventh, eliminating the diminished fifth that Alpha Prime produces.

*****

EXAMPLE 74:

Listen to Example 74

If we perform the same exercise with pure minor, which is the Alpha 6 or Aeolian mode, instead of a series of major/major sevenths acting as subdominants, we get a series of minor/minor sevenths. Again, the secondary subdominants are a different species of harmony than the tonic, as they are Alpha 2/Dorian sonorities.

Paleo-theorists handled the minor modes differently with respect to where the secondary dominants reside, but through the examples here we see that the only real difference between major and minor is what gender the primary dominant resolves to. I resolved to a final major tonic to accentuate this, which is the ancient Tierce de Picardie practice that goes back as far as composers have been resolving to triads instead of open fifths: Musical intuition is a powerful force.

Note again that the transit through the secondary subdominant sub-system eliminates the diminished fifth in the Alpha Prime version of the V(m7)/iii.

*****

EXAMPLE 75:

Listen to Example 75

The Beta Prime contextual system showed us that a subdominant could be an overtone chord too, and with that we end up with a full chromatic cycle of overtone sonorities. Omega is a good name for this system, as, "beyond here there be dragons" so to speak. It is easy to get totally lost as to where you are without following the music or at least listening closely, and this will bring up the ultimate implication of the overtone sonorities resolutional paradigm in the following example.

I again ended with a Picardy third to imply that the tonic of this system could also be major.

*****

This is what is inescapably ingrained into us by the overtone series resolutional paradigm: An endless series of overtone sonorities resolving to major triads, which in turn acquire minor sevenths to become new overtone chords and continue the progressive cycle. If you have ever lived anywhere there are maple trees, it is an auditory analogue to watching a maple seed fall, spiraling to the ground with its gyroscopic spinning: I can land on a rock or on the ground, just as any of the following overtone sonorities can alight on a major or a minor tonic.

I have only provided a two octave cycle, but subconsciously this pattern fills the entire spectrum of human hearing.

EXAMPLE 76A:

EXAMPLE 76B:

Listen to Example 76

Sorry for the Bizarro World piano sound font, but I neglected to properly assign the PC88 Marcato Strings to that example. Arg.

*****

There are many, many possibilities for fully integrated contextual systems - which I think I'll put into an appendix in the final book - but I just want to demonstrate one that has all twenty-four major and minor tonics as well as the twelve overtone chords. Each tonic starts minor, becomes major, then acquires a major seventh, and finally a minor seventh to become an overtone sonority. Obviously - if you've been following to this point - this creates a strict double canon.

EXAMPLE 77:

Listen to Example 77

This could be further adorned with diminished fifths and minor ninths of course, but I really like this one.

That brings to a conclusion the section of WMW that deals with harmony. There are sections to follow on counterpoint, rhythm, and form, but I don't know when I'll get to them, because I have other projects I need to get to right now. In fact, I've been anxious to finish this up because of some things I'm getting to the critical mass point with, the primary one of which is my programming of the Yamaha FS1R FM/Formant Sequence digital synthesizer.

Stay tuned.

Why Music Works: Chapter Fourteen

2010-10-16T15:27:00.001-05:00

The Pythagorean Comma and Equal Temperament

PREFACE to All Posts:

This is to be the culmination of the Musical Relativity series of posts I did back in 2006, which can be found to your right in the sidebar. Back then I was calling the series Musical Implications of the Harmonic Overtone Series. Even before that, I did a series of posts called Harmonic Implications of the Overtone Series that started this all. Here, I am presenting the final weblog version of the evolving book I've decided to publish with the intention of getting some feedback before I create the final print version, which I plan to put into the ePub format for iBooks. So, please feel free to ask any questions about anything that you think I haven't made perfectly clear, and don't hesitate to offer any constructive criticisms or suggestions. Since this project is the accidental result of several decades of curios inquiry - and many prominent and also relatively anonymous theorists and teachers have contributed ideas to it (Which I will credit where memory serves and honor dictates) - I am eager to get a final layer of polish from any and all who may happen to read this series and find it useful, or potentially so. Since I am creating these posts as .txt files first, revision should be a simple process.

Since my pre-degree studies at The Guitar Institute of the Southwest and my undergraduate work at Berklee College of Music looked at music theory from the jazz perspective, and then my master of music and doctor of musical arts studies at Texas State and The University of North Texas were from the traditional perspective, a large part of how I discovered the things in this book-in-progress was the result of my trying to reconcile those different theoretical viewpoints. Since I want this work to be of practical value, I have retained all of the traditional theoretical nomenclature possible, and only added to it where necessary to describe phenomena that have not heretofore been present in musical analysis. I have, however, standardized terminology into what I think is the most logical system yet devised, and that will be explained as the reader goes along. There is a lot of built-in review and repetition - something I've learned from my decades of private teaching - so even a once-through with this systematic approach to understanding musical phenomena ought to be of significant benefit.

Outright addition to traditional musical analysis is limited to the symbology required to label root motion and transformation types so that the root motion and transformation patterns are visible: This greatly facilitates comprehension, and since good and bad harmonic continuities are separated by the effectiveness or lack thereof in the root motion and transformation patterns, this also actually functions as an aid to composition. All symbology - old and new - has been worked out over the past three decades so that everything is readily available with the standard letters, numbers, and symbols found on a QWERTY keyboard.

Finally, for the contextual systems, I have used the Greek alphabet: The normal diatonic systems - those comprised of two minor seconds and five major seconds - are Alpha, Beta, and Gamma. The exotic diatonic systems - those that contain a single augmented second - are Delta, Epsilon, and Zeta, and finally, the alien diatonic systems - those that contain two augmented seconds - are Eta, Theta, and Iota. Since the theoretical writings that started western art music out were handed down from ancient Greece, I thought this would be a fitting tribute, as well as a handy and logical classification scheme.

Basically, if you have a baccalaureate-level understanding of music theory from either a jazz or traditional perspective, you should have no problem understanding anything in this straight-forward treatise.

INTRODUCTION to Chapter Fourteen:

In chapter one, I demonstrated how the overtone sonority generates the three normal diatonic systems - those seven note systems that contain two semitones and five tones: Alpha, Beta, and Gamma - and then in chapter two we examined each of those systems in detail, discovering that the primacy of Alpha is due to the fact that all seven of its harmonies can be arranged in progressive order. In chapter three, we examined the contextualization of Alpha Prime, looking at the various different root progressions types it can exhibit, their various transformations, and through this we also started to look into the world of musical effect and affect. Chapter four was dedicated to examining how Beta Prime and Gamma Prime compared to Alpha, using the same musical proof formats developed in chapter three. Through those proofs, we discovered some very unusual harmonic effects that evoke the uncanny that are contained in the Beta and Gamma systems. Chapter five then took us out of the diatonic harmonic world and into the chromatic realm as we discovered the origins of the secondary dominant sub-system sonorities. After the secondary dominants, in chapter six, we looked at the secondary subdominant sub-system of harmonies, which completed a larger set of integrated chromatic systems, which we will look at in detail later.

Then in chapter seven, we looked at the exotic diatonic systems - those seven note contextual systems that contain a single augmented second: Delta, Epsilon, and Zeta - and in chapter eight we looked in detail at the root motion types they contain, and the unique harmonic effects that these unusual systems create. With the exotic systems out of the way, in chapter nine, I was free to demonstrate a phenomenon that is an artifact of patterned root progressions, which I pointed out earlier, and that is harmonic canon. Depending upon how harmonic canons are developed and set up, I showed how they can also exhibit the phenomena I call Musical Escher Morphs and Harmonic Mobius Loops. Returning to diatonic contextual systems in chapter ten, I introduced the alien diatonic systems - which are those seven note systems that contain two augmented seconds: Eta, Theta, and Iota - and then its companion, chapter eleven, examined the isolated root motion and transformation types in those alien systems.

After finishing analysis of the nine diatonic contextual systems with the comparative morphology example in chapter twelve, we then turned to the extra-diatonic contextual systems of Kappa, Lambda, and Mu in chapter fourteen. Now it's time to look at the series generated intervals, the Pythagorean Comma, and the inevitability of equal temperament for fixed pitch instruments.

*****

CHAPTER FOURTEEN:

EXAMPLE 72:

Listen to Example 69

The simplest way to wrap your brain around the Pythagorean comma and equal temperament is the illustration in example seventy-two above. If we stack seven perfect 2:1 octaves next to twelve perfect 3:2 fifths - both starting on F-natural - we should end up on the same pitch, which is the TTET-equivalent F-natural/E-sharp. The problem is that with twelve pure 3:2 perfect fifths, this doesn't happen: The E-sharp is 23.46 cents above the F-natural. In TTET, a cent is 1/100 - or 1% - of an equally tempered semitone, so the E-sharp is slightly less than eighth-tone sharp. Since average ears can detect five to seven cents of difference and well trained ears two or three cents, this is obviously out of tune.

Since the octave is inviolable - pitch recognition depends on octave equivalence (Though piano tuners mess with them all the time) - the simplest and most elegant solution is to simply reduce the size of the perfect fifths by one twelfth of the comma, which comes out to 1.955 cents, or less than 2% of a semitone. I can hear this with two simultaneous notes a fifth apart in TTET (So long as there are no phase, flange, chorus, or reverb effects present) as a slow cross-modulation - or beating - but I understand that the average listener doesn't perceive that level of detail. This trivial adjustment would seem to completely cure the "problem," but it doesn't really, since the other ratios that the harmonic series generates do not equally fill the perfect fifth (A just major and minor third, for example).

EXAMPLE 72B:

There is a useful chart demonstrating the differences between just and equally tempered intervals at the Wikipedia article about equal temperament.

As you can see, while the deviations from pure for the perfect fifth and perfect fourth are indeed trivial, that is not the case for some of the other intervals. What is true, however, is that all of the deviations from pure in equal temperament are trivial for the perfect consonant intervals. This really is the most important thing, because those are the lowest - and therefore most prominent - overtones.

For the imperfect consonances, there are some seeming problems, but perceptually some of those seeming problems are not as bad as they look. Almost no average listener has any problems with minor thirds and major sixths in TTET, for example, but I and some others do perceive them as slightly "off," but still not bothersome. In contrast to that, the major thirds and minor sixths do bother some people, including me, even though their deviations are smaller than those for the minor third/major sixth pair. Again, that is because the major third is lower in the series, more prominent as an overtone, and the overtone chord is based on it as a lowest gender-defining interval. As a result of the strident nature of equally tempered thirds, I and some other guitarists detune our G strings because in guitar music so many major thirds occur between the D and G strings. Luthiers often compensate saddles there too.

As for the dissonances, there is so much interference in them already because of the complexities involved in their relationships that it really doesn't matter much, with one exception: The tritone. In just tuning an augmented fourth and a diminished fifth are not the same interval: The diminished fifth is sharp compared to an equally tempered tritone, and an augmented fourth is flat compared to an equally tempered tritone (I use tritone to describe both augmented fourths and diminished fifths in TTET because they are the same 600 cent interval). That means when passive tones are present to define those intervals in their respective ways, the pure diminished fifth does not have as strong of a leading tone/leaning tone impetus to contract, and the pure augmented fourth doesn't have as strong of a leaning tone/leading tone impetus to expand.

This means that, as far as I am concerned, an equally tempered tritone is superior to the two just versions. The naturally just overtone sonority already wishes to resolve, and TTET only adds to that desire.

*****

I found an interesting YouTube video that purports to demonstrate the superiority of root-calculated just harmony, but which to me in fact proves just the opposite: The superiority of equal temperament. The two things that make this example so excellent are that the sound is generated as sine waves, and that there is a wrapped oscilloscope for visual reenforcement.

What I think is one of the most humorous aspects to that video is the fact that any critical thinker will immediately notice that you can see a lot more difference than you can hear.

*****

Another way to view the naturalness of the equal temperament solution is by noting that the harmonic series is made up of simple ratios generated beyond the octave while equal temperament is made up primarily of simple ratios plotted within the octave. For example, in equal temperament the octave is 1, the tritone is 1/2, the major third is 1/3, the minor third is 1/4, a whole step is 1/6, and a semitone is 1/12. The perfect fourth and fifth, which are 5/12 and 7/12 respectively, are really the only complex intervals in the system, and those are the ones closest to just. Noting TTET variations from just isn't exactly a making a red herring argument, but it's close.

A final thing to think about is what kinds of effects we apply to music today. I previously mentioned phasing and flanging, which are short time delay effects with small pitch modulations that increase disturbances to solid perception of pitch but it is really the ubiquitous use of pitch-shift and delay/pitch mod chorus that puts the lie to the notion that people want to hear just harmony: Just the opposite is true, and it has been since people first noticed how nice a section of doubled or trebled voices or strings sounded. And voices and strings can theoretically reproduce just intervals and harmony.

I enjoy hearing historical temperaments for period music, but I want all twenty-four major and minor keys to sound the same relative to each other, and TTET is the only way to get that with fixed pitch instruments, so it is the only viable solution as far as I am concerned.

That leads us to the final chapter in the current version of WMW on harmony.

Why Music Works: Chapter Thirteen

2010-10-13T09:27:00.000-05:00

Genesis of the Extra-Diatonic Contextual Systems: Kappa, Lambda, and Mu

PREFACE to All Posts:

This is to be the culmination of the Musical Relativity series of posts I did back in 2006, which can be found to your right in the sidebar. Back then I was calling the series Musical Implications of the Harmonic Overtone Series. Even before that, I did a series of posts called Harmonic Implications of the Overtone Series that started this all. Here, I am presenting the final weblog version of the evolving book I've decided to publish with the intention of getting some feedback before I create the final print version, which I plan to put into the ePub format for iBooks. So, please feel free to ask any questions about anything that you think I haven't made perfectly clear, and don't hesitate to offer any constructive criticisms or suggestions. Since this project is the accidental result of several decades of curios inquiry - and many prominent and also relatively anonymous theorists and teachers have contributed ideas to it (Which I will credit where memory serves and honor dictates) - I am eager to get a final layer of polish from any and all who may happen to read this series and find it useful, or potentially so. Since I am creating these posts as .txt files first, revision should be a simple process.

Since my pre-degree studies at The Guitar Institute of the Southwest and my undergraduate work at Berklee College of Music looked at music theory from the jazz perspective, and then my master of music and doctor of musical arts studies at Texas State and The University of North Texas were from the traditional perspective, a large part of how I discovered the things in this book-in-progress was the result of my trying to reconcile those different theoretical viewpoints. Since I want this work to be of practical value, I have retained all of the traditional theoretical nomenclature possible, and only added to it where necessary to describe phenomena that have not heretofore been present in musical analysis. I have, however, standardized terminology into what I think is the most logical system yet devised, and that will be explained as the reader goes along. There is a lot of built-in review and repetition - something I've learned from my decades of private teaching - so even a once-through with this systematic approach to understanding musical phenomena ought to be of significant benefit.

Outright addition to traditional musical analysis is limited to the symbology required to label root motion and transformation types so that the root motion and transformation patterns are visible: This greatly facilitates comprehension, and since good and bad harmonic continuities are separated by the effectiveness or lack thereof in the root motion and transformation patterns, this also actually functions as an aid to composition. All symbology - old and new - has been worked out over the past three decades so that everything is readily available with the standard letters, numbers, and symbols found on a QWERTY keyboard.

Finally, for the contextual systems, I have used the Greek alphabet: The normal diatonic systems - those comprised of two minor seconds and five major seconds - are Alpha, Beta, and Gamma. The exotic diatonic systems - those that contain a single augmented second - are Delta, Epsilon, and Zeta, and finally, the alien diatonic systems - those that contain two augmented seconds - are Eta, Theta, and Iota. Since the theoretical writings that started western art music out were handed down from ancient Greece, I thought this would be a fitting tribute, as well as a handy and logical classification scheme.

Basically, if you have a baccalaureate-level understanding of music theory from either a jazz or traditional perspective, you should have no problem understanding anything in this straight-forward treatise.

INTRODUCTION to Chapter Thirteen:

In chapter one, I demonstrated how the overtone sonority generates the three normal diatonic systems - those seven note systems that contain two semitones and five tones: Alpha, Beta, and Gamma - and then in chapter two we examined each of those systems in detail, discovering that the primacy of Alpha is due to the fact that all seven of its harmonies can be arranged in progressive order. In chapter three, we examined the contextualization of Alpha Prime, looking at the various different root progressions types it can exhibit, their various transformations, and through this we also started to look into the world of musical effect and affect. Chapter four was dedicated to examining how Beta Prime and Gamma Prime compared to Alpha, using the same musical proof formats developed in chapter three. Through those proofs, we discovered some very unusual harmonic effects that evoke the uncanny that are contained in the Beta and Gamma systems. Chapter five then took us out of the diatonic harmonic world and into the chromatic realm as we discovered the origins of the secondary dominant sub-system sonorities. After the secondary dominants, in chapter six, we looked at the secondary subdominant sub-system of harmonies, which completed a larger set of integrated chromatic systems, which we will look at in detail later.

Then in chapter seven, we looked at the exotic diatonic systems - those seven note contextual systems that contain a single augmented second: Delta, Epsilon, and Zeta - and in chapter eight we looked in detail at the root motion types they contain, and the unique harmonic effects that these unusual systems create. With the exotic systems out of the way, in chapter nine, I was free to demonstrate a phenomenon that is an artifact of patterned root progressions, which I pointed out earlier, and that is harmonic canon. Depending upon how harmonic canons are developed and set up, I showed how they can also exhibit the phenomena I call Musical Escher Morphs and Harmonic Mobius Loops. Returning to diatonic contextual systems in chapter ten, I introduced the alien diatonic systems - which are those seven note systems that contain two augmented seconds: Eta, Theta, and Iota - and then its companion, chapter eleven, examined the isolated root motion and transformation types in those alien systems.

After finishing analysis of the nine diatonic contextual systems with the comparative morphology example in chapter twelve, we now turn to the extra-diatonic contextual systems of Kappa, Lambda, and Mu. Extra-diatonic systems are those that lie between the seven note diatonic systems and the fully chromatic twelve note systems. I have mentioned one of these nonatonic - nine note - systems previously, and that is melodic minor, which will start our examples off in the present chapter.

*****

CHAPTER THIRTEEN:

EXAMPLE 69:

Listen to Example 69

As I mentioned way back in an early chapter, the Kappa contextual system is easily internalized as a bi-modal combination of Alpha 6 - the Aeolian mode - and Beta Prime, which is Aeolian with a major sixth and seventh. Traditionally, Beta Prime has been used as the ascending form, and Alpha 6 for the descending. These conventions were not, however, scrupulously followed in common practice music, as a descending line over a IV(m7) or V(m7) harmony required the ascending form and its attendant raised degrees.

That leads to the true way that nature tells us this system is generated, which begins with an overtone sonority resolving to a minor/minor seventh tonic, as we see in the first progressive resolution above. This yields fully seven of the nine required notes all on its own, since we are no longer hemmed in by the diatonic system's paradigm of retaining the inflection of notes that appeared in a previous harmony: The B-flat minor seventh over the tonic is now perfectly acceptable (And it sounds less hotly dissonant than the minor/major seventh as well).

With the seven note diatonic systems, all that was required was a single additional progressive resolution to the subdominant, and the system was completely generated. With Kappa Prime, however, the resolution to IV(m7) only adds the eighth note - A-natural - so an additional third progressive resolution to the secondary subdominant of bVII(m7) is needed to get the A-flat. At this point, the astute reader will realize that there is no reason to stop the progressive resolutions there, and that fully chromatic systems are also implied by nature's resolutional paradigm. We will examine the manifold possibilities of chromatic contextual systems at a later point.

*****

EXAMPLE 70:

NOTE: The example below has an error in it, as the primary subdominant should be an overtone chord. Unfortunately, I didn't discover that until I had uploaded everything, so I'll have to fix that for the final book. One of the points of doing this series of posts is to iron that stuff out.

Listen to Example 70

The Lambda contextual system is commonly heard in Flemenco music, and the simplest way for traditionally trained musicians to internalize it is as a bi-modal combination of Alpha 3 - the Phrygian mode - and Alpha 5 which is Mixolydian. In vernacular usage, the v(d5m7) dominant stand-in is seldom heard, but the bII(M7) subdominant function harmony is (Usually as a triad). Though nominally a decatonic - ten note - system, the singers and soloists - usually guitarists - adhere pretty regularly to the Phrygian mode for vocals and improvisation.

For the first progressive resolution here we have the previously mentioned v(d5m7) - which is native to Alpha 3/Phrygian - resolving to an overtone sonority on the tonic degree. This yields six of the ten required notes, and the further resolution to the IV(m7) subdominant - which is usually replaced by the Phrygian-native minor/minor seventh in this idiom - gets the system to nine notes. So once again, a further resolution into the secondary subdominant realm is required to pick up the final note, which is the D-natural that is native to Alpha 5/Mixolydian. As I stated earlier, in the idiomatic traditional employment of this system, the vast majority of the melodic elements adhere to the diatonic Alpha 3/Phrygian mode.

The acoustical reason that this bi-modal combination works is because the dissonant overtone sonority can support a wide variety of upper structure tones, such as the minor ninth, augmented ninth/minor third, and the minor thirteenth that Phrygian adds to Mixolydian here. This same ability of the overtone sonority to support a wide variety of tones in the upper structures is also responsible for our final Mu contextual system.

*****

EXAMPLE 71:

Listen to Example 71

The Mu contextual system is the modern king of all of the extra-diatonic systems, as it virtually defined the sound of the twentieth century through it's ubiquitous employment in blues music, which evolutionarily lead to jazz, R&B, and rock and roll. In fact, I start all of my students out with the blues - even if they wish to study classical music - for this very reason: You can't understand the music of the twentieth century without a solid foundation in the blues.

For the traditionally trained theorists among us, the easiest way to pocket an understanding of blues tonality is as a tri-modal combination of Alpha 2 - the Dorian mode - Alpha 5/Mixolydian, and Alpha Prime. In the idiomatic vernacular, what you see above is exactly what you get: Overtone sonorities on all three cardinal degrees. In fact, in the most basic blues and blues based R&B and rock and roll music, these are the only three chords you ever hear. By the time swing and bebop guys like Charlie Parker got ahold of the blues, however, it was elevated into another musical art form entirely, with many harmonic excursions within its brief twelve measure form.

Obviously, the overtone sonorities on the dominant, tonic, and subdominant correspond to Ionian, Mixolydian, and Dorian respectively. In idiomatic practice, blues, rhythm and blues, blues based rock, and blues based jazz singers and soloists employ a multi-layered combination of modalities, which consist of a minor pentatonic skeleton, the fleshed out diatonic Dorian, and then a virtually fully clothed chromatic sub-system of passing and approach notes for the bling. Blues is something a beginner can learn to play in a week, and then explore for a lifetime, which is the primary aspect of its charm and enduring appeal.

Why Music Works: Chapter Twelve

2010-10-10T04:27:00.001-05:00

Harmonic Mobius Loops, Harmonic Palindromes, and Comparative Morphology of the Diatonic Contextual Systems

PREFACE to All Posts:

This is to be the culmination of the Musical Relativity series of posts I did back in 2006, which can be found to your right in the sidebar. Back then I was calling the series Musical Implications of the Harmonic Overtone Series. Even before that, I did a series of posts called Harmonic Implications of the Overtone Series that started this all. Here, I am presenting the final weblog version of the evolving book I've decided to publish with the intention of getting some feedback before I create the final print version, which I plan to put into the ePub format for iBooks. So, please feel free to ask any questions about anything that you think I haven't made perfectly clear, and don't hesitate to offer any constructive criticisms or suggestions. Since this project is the accidental result of several decades of curios inquiry - and many prominent and also relatively anonymous theorists and teachers have contributed ideas to it (Which I will credit where memory serves and honor dictates) - I am eager to get a final layer of polish from any and all who may happen to read this series and find it useful, or potentially so. Since I am creating these posts as .txt files first, revision should be a simple process.

Since my pre-degree studies at The Guitar Institute of the Southwest and my undergraduate work at Berklee College of Music looked at music theory from the jazz perspective, and then my master of music and doctor of musical arts studies at Texas State and The University of North Texas were from the traditional perspective, a large part of how I discovered the things in this book-in-progress was the result of my trying to reconcile those different theoretical viewpoints. Since I want this work to be of practical value, I have retained all of the traditional theoretical nomenclature possible, and only added to it where necessary to describe phenomena that have not heretofore been present in musical analysis. I have, however, standardized terminology into what I think is the most logical system yet devised, and that will be explained as the reader goes along. There is a lot of built-in review and repetition - something I've learned from my decades of private teaching - so even a once-through with this systematic approach to understanding musical phenomena ought to be of significant benefit.

Outright addition to traditional musical analysis is limited to the symbology required to label root motion and transformation types so that the root motion and transformation patterns are visible: This greatly facilitates comprehension, and since good and bad harmonic continuities are separated by the effectiveness or lack thereof in the root motion and transformation patterns, this also actually functions as an aid to composition. All symbology - old and new - has been worked out over the past three decades so that everything is readily available with the standard letters, numbers, and symbols found on a QWERTY keyboard.

Finally, for the contextual systems, I have used the Greek alphabet: The normal diatonic systems - those comprised of two minor seconds and five major seconds - are Alpha, Beta, and Gamma. The exotic diatonic systems - those that contain a single augmented second - are Delta, Epsilon, and Zeta, and finally, the alien diatonic systems - those that contain two augmented seconds - are Eta, Theta, and Iota. Since the theoretical writings that started western art music out were handed down from ancient Greece, I thought this would be a fitting tribute, as well as a handy and logical classification scheme.

Basically, if you have a baccalaureate-level understanding of music theory from either a jazz or traditional perspective, you should have no problem understanding anything in this straight-forward treatise.

INTRODUCTION to Chapter Eleven:

In chapter one, I demonstrated how the overtone sonority generates the three normal diatonic systems - those seven note systems that contain two semitones and five tones: Alpha, Beta, and Gamma - and then in chapter two we examined each of those systems in detail, discovering that the primacy of Alpha is due to the fact that all seven of its harmonies can be arranged in progressive order. In chapter three, we examined the contextualization of Alpha Prime, looking at the various different root progressions types it can exhibit, their various transformations, and through this we also started to look into the world of musical effect and affect. Chapter four was dedicated to examining how Beta Prime and Gamma Prime compared to Alpha, using the same musical proof formats developed in chapter three. Through those proofs, we discovered some very unusual harmonic effects that evoke the uncanny that are contained in the Beta and Gamma systems. Chapter five then took us out of the diatonic harmonic world and into the chromatic realm as we discovered the origins of the secondary dominant sub-system sonorities. After the secondary dominants, in chapter six, we looked at the secondary subdominant sub-system of harmonies, which completed a larger set of integrated chromatic systems, which we will look at in detail later.

Then in chapter seven, we looked at the exotic diatonic systems - those seven note contextual systems that contain a single augmented second: Delta, Epsilon, and Zeta - and in chapter eight we looked in detail at the root motion types they contain, and the unique harmonic effects that these unusual systems create. With the exotic systems out of the way, in chapter nine, I was free to demonstrate a phenomenon that is an artifact of patterned root progressions, which I pointed out earlier, and that is harmonic canon. Depending upon how harmonic canons are developed and set up, I showed how they can also exhibit the phenomena I call Musical Escher Morphs and Harmonic Mobius Loops. Returning to diatonic contextual systems in chapter ten, I introduced the alien diatonic systems - which are those seven note systems that contain two augmented seconds: Eta, Theta, and Iota - and then its companion, chapter eleven, examined the isolated root motion and transformation types in those alien systems.

Now in chapter twelve, we return to the phenomenon I mentioned in chapter nine called harmonic Mobius loops, and the closely related phenomenon of harmonic palindromes.

*****

CHAPTER TWELVE:

EXAMPLE 67:

Listen to Example 67

An harmonic Mobius loop is an harmonic continuity in which the root motion types are balanced out so that the ending voicing leads back into the beginning voicing. Almost all composers, working intuitively, will create continuities in which the voices descend over time, as the natural, ingrained tendency that intuitive understanding of the overtone sonority's resolutional desire imparts, is to use more progressive type motions: Progressions, half-progressions, and super-progressions. These progressive types of root motion are statistically more common in all forms of harmonic music for this reason, and that is what lead Schenker to devise his theories, which he didn't understand were only observations of an artifact of music that naturally contained a preponderance of these progressive type root motions. So, it takes both understanding and discipline to create harmonic Mobius loops, as the natural tendency is against creating them.

The easiest way to construct a harmonic Mobius loop is actually to create a harmonic palindrome, because the forward and reverse root motions will balance each other out so that the ending voicing leads back to the beginning voicing, as is desired. I have demonstrated this in the first part of example sixty-seven above.

To begin construction of one of these symmetrical Mobius loops - another way to describe an harmonic palindrome - is to put the tonic harmony in the first measure and the dominant harmony in the last measure. Since the dominant to tonic resolution is a progression, that means measure two has to be a regression back to the dominant. After that, the next three root motions/harmonies are up to the composer. Obviously, this limits the possibilities, especially in a single eight measure phrase, as we're working with here. I chose to go vi(m7), IV(M7), to ii(m7), as that is one way to yield the maximum possible five different harmonies in eight measures.

Breaking away from the symmetry imposed by harmonic palindromes can actually yield more interesting harmonic Mobius loops, as I have demonstrated in the second part of example sixty-seven. All I did here, obviously, was to lower the harmonies a step within the diatonic system for measures two, three, and four: The final four measures are the same. The reason this works here is because the initial progressive root motion is now balanced out by a super-regressive root motion into measure five: Note that the voicing in the fifth measure of each example is identical. This also has the additional benefit of presenting six of the diatonic harmonies instead of only five.

To present a comparative morphology of the nine primary diatonic systems, however, I will be required to present all seven diatonic harmonies in the Mobius loop. Therefore, neither of these two examples will suffice. I also would like to present as many different root motion types as possible - ideally all eight of them: Progression, regression, half-progression, half-regression, super-progression, super-regression, progressive tritone, and regressive tritone - so this will call for a more artful approach.

EXAMPLE 68A:

NOTE: I will present the audio example link at the end of the variation set.

My final harmonic Mobius loop starts out as the first part of sixty-seven did, but to get a progressive tritone in, I went from IV(M7) to vii(d5m7) from measure four into measure five. I then reversed the progression types in the first four measures, as you can see, in an attempt to balance things out and also to get all seven harmonies in: Progression answers regression, super-regression answers super-progression, and half-regression answers half-progression. I did manage to get all seven harmonies in - which was the primary goal - and I got seven out of eight root motion types in (Only the regressive tritone is missing) - which was the secondary goal, but I did not quite manage to balance the root motion types out, as there is still a preponderance of progressive types if we super-progress into the dominant in measure eight. If I went directly from the subdominant to the tonic, it would work, but that does not produce the desired dominant resolution, and if I went from the subdominant directly to the dominant, the voicings would miss their loop by a single inversion (7, 1, 3, 5 instead of the required 1,3,5, 7). So, I had to add another half-regression to make the loop happen. Doing this creates a nice turnaround figure that will alert the listener that a new variation and diatonic contextual system is forthcoming - and the final measure is a rhythmically diminished retrograde of the second three measures - so that makes this harmonic Mobius loop ideal for presenting a comparative morphology of the nine diatonic contextual systems.

Since this is a systematic run-through of only the nine master contexts, there are actually many nice possibilities within the sub-contexts not presented here. Presenting even only the independent sub-contexts, however, would lead to a huge and ponderous example: This is more than sufficient to alert the musically aware listener of the manifold new resources available, many of which remain almost totally unexplored.

I presented Alpha Prime twice here to allow the listener to get the lay of the land. I will not hold the reader's hand through this with a detailed explanation of every effect, as I assume that those for whom I created this will possess at least the basic intellectual curiosity to see that everything is in the analysis on the example pages.

EXAMPLE 68B:

This page presents the other two Native Diatonic Systems with Beta Prime and Gamma Prime. Since the only difference now is the presence of E-flat, there are both familiar and peculiar effects present. The last quarter note of measure twenty-four carries a V(d5m7) sonority, as this is the altered dominant belonging to the Gamma Prime contextual system: I anticipate the upcoming systems by leading into them with their dominants and dominant stand-ins throughout the variation set.

Gamma Prime, then, adds a D-flat to the E-flat already present, so the harmonic effects are more strange.

EXAMPLE 68C:

Page three gets us into the Exotic Diatonic Contextual Systems with Delta Prime, so our tonic is again a major/major seventh sonority. A-flat is the only disturbance to Alpha Prime here, so again there are both familiar and unfamiliar effects present. Epsilon Prime adds an E-flat, though, so we're back to a minor/major seventh tonic, and there are more unusual effects present.

EXAMPLE 68D:

Zeta Prime is the last of the Exotic Diatonic Contextual Systems, and it adds a D-flat to the A-flat and E-flat previously present: Now, all of the harmonies are strange compared to Alpha Prime.

With Eta Prime, we enter the Alien Diatonic Contextual Systems, and though the tonic is again a major/major seventh, the presence of the two augmented seconds produces copious amounts of uncanniness. The appearance of the V(d5M7) dominant stand-in - with it's F-sharp - really destabilizes everything, though.

EXAMPLE 68E:

The addition of the F-sharp with Theta Prime means that the final measure of the Mobius structure now has double chromatic approach tones to the dominant stand-in, which is quite nice. It's even better into a real dominant, as we see and hear in the final measure on the page where, finished with the nine contextual systems, we prepare to return to Alpha Prime. That brings up a point: Not only can you explore the sub-contexts as you please, but also nothing is stopping you from mixing and matching the contexts and sub-contexts to get whatever effects you desire on whatever degree you desire them. The possibilities available through a complete understanding of the diatonic contextual systems are truly staggering, and as I mentioned, many of them remain virtually unexplored, yet easily accessible.

EXAMPLE 68F:

Listen to Example 68

And with that, we return to Alpha Prime, where I again provide two statements of the Mobius loop to reorient yourselves, and also an ending, which finally allows the B-natural to resolve up to C-natural.

After listening to the example, you can hear that, no matter how bizarre the harmonic effects get, it is not only still possible to recognize the theme, it is impossible not to recognize it. This is one area where the so-called atonal composers of the twentieth century utterly failed: Variations on atonal constructs are virtually impossible to recognize, let alone follow, for the lay listener. Even for those of us with acutely well trained ears, it isn't exactly a satisfying experience, never mind fun.

The presence of a viable contextual system, no matter how warped, solves this problem completely, and as I said, the sub-contexts not presented here contain many more points of interest.

Why Music Works: Chapter Eleven

2010-10-07T02:27:00.000-05:00

Root Motion and Transformation Types in the Eta, Theta, and Iota Systems

PREFACE to All Posts:

This is to be the culmination of the Musical Relativity series of posts I did back in 2006, which can be found to your right in the sidebar. Back then I was calling the series Musical Implications of the Harmonic Overtone Series. Even before that, I did a series of posts called Harmonic Implications of the Overtone Series that started this all. Here, I am presenting the final weblog version of the evolving book I've decided to publish with the intention of getting some feedback before I create the final print version, which I plan to put into the ePub format for iBooks. So, please feel free to ask any questions about anything that you think I haven't made perfectly clear, and don't hesitate to offer any constructive criticisms or suggestions. Since this project is the accidental result of several decades of curios inquiry - and many prominent and also relatively anonymous theorists and teachers have contributed ideas to it (Which I will credit where memory serves and honor dictates) - I am eager to get a final layer of polish from any and all who may happen to read this series and find it useful, or potentially so. Since I am creating these posts as .txt files first, revision should be a simple process.

Since my pre-degree studies at The Guitar Institute of the Southwest and my undergraduate work at Berklee College of Music looked at music theory from the jazz perspective, and then my master of music and doctor of musical arts studies at Texas State and The University of North Texas were from the traditional perspective, a large part of how I discovered the things in this book-in-progress was the result of my trying to reconcile those different theoretical viewpoints. Since I want this work to be of practical value, I have retained all of the traditional theoretical nomenclature possible, and only added to it where necessary to describe phenomena that have not heretofore been present in musical analysis. I have, however, standardized terminology into what I think is the most logical system yet devised, and that will be explained as the reader goes along. There is a lot of built-in review and repetition - something I've learned from my decades of private teaching - so even a once-through with this systematic approach to understanding musical phenomena ought to be of significant benefit.

Outright addition to traditional musical analysis is limited to the symbology required to label root motion and transformation types so that the root motion and transformation patterns are visible: This greatly facilitates comprehension, and since good and bad harmonic continuities are separated by the effectiveness or lack thereof in the root motion and transformation patterns, this also actually functions as an aid to composition. All symbology - old and new - has been worked out over the past three decades so that everything is readily available with the standard letters, numbers, and symbols found on a QWERTY keyboard.

Finally, for the contextual systems, I have used the Greek alphabet: The normal diatonic systems - those comprised of two minor seconds and five major seconds - are Alpha, Beta, and Gamma. The exotic diatonic systems - those that contain a single augmented second - are Delta, Epsilon, and Zeta, and finally, the alien diatonic systems - those that contain two augmented seconds - are Eta, Theta, and Iota. Since the theoretical writings that started western art music out were handed down from ancient Greece, I thought this would be a fitting tribute, as well as a handy and logical classification scheme.

Basically, if you have a baccalaureate-level understanding of music theory from either a jazz or traditional perspective, you should have no problem understanding anything in this straight-forward treatise.

INTRODUCTION to Chapter Eleven:

In chapter one, I demonstrated how the overtone sonority generates the three normal diatonic systems - those seven note systems that contain two semitones and five tones: Alpha, Beta, and Gamma - and then in chapter two we examined each of those systems in detail, discovering that the primacy of Alpha is due to the fact that all seven of its harmonies can be arranged in progressive order. In chapter three, we examined the contextualization of Alpha Prime, looking at the various different root progressions types it can exhibit, their various transformations, and through this we also started to look into the world of musical effect and affect. Chapter four was dedicated to examining how Beta Prime and Gamma Prime compared to Alpha, using the same musical proof formats developed in chapter three. Through those proofs, we discovered some very unusual harmonic effects that evoke the uncanny that are contained in the Beta and Gamma systems. Chapter five then took us out of the diatonic harmonic world and into the chromatic realm as we discovered the origins of the secondary dominant sub-system sonorities. After the secondary dominants, in chapter six, we looked at the secondary subdominant sub-system of harmonies, which completed a larger set of integrated chromatic systems, which we will look at in detail later.

Then in chapter seven, we looked at the exotic diatonic systems - those seven note contextual systems that contain a single augmented second: Delta, Epsilon, and Zeta - and in chapter eight we looked in detail at the root motion types they contain, and the unique harmonic effects that these unusual systems create. With the exotic systems out of the way, in chapter nine, I was free to demonstrate a phenomenon that is an artifact of patterned root progressions, which I pointed out earlier, and that is harmonic canon. Depending upon how harmonic canons are developed and set up, I showed how they can also exhibit the phenomena I call Musical Escher Morphs and Harmonic Mobius Loops.

In the previous chapter, chapter ten, I introduced the alien diatonic systems - which are those seven note systems that contain two augmented seconds: Eta, Theta, and Iota - and with today's chapter eleven we will finish examining all of the nine master diatonic contextual systems and the total of sixty-three independent and dependent diatonic modes. All that is left is to look at and listen to the isolated root motion and transformation types.

*****

CHAPTER ELEVEN:

EXAMPLE 58:

Listen to Example 58

Here we have the progressions and regressions in Eta Prime, and you may find the effects other-worldly, as I do, which is what lead me to classify these contextual systems as alien: They are very, very far removed from the native systems, and even more foreign than the exotic systems.

EXAMPLE 59:

Listen to Example 59

These relatively smoother half-progressions and half-regressions can't do much to mitigate the strange harmonic effects that Eta Prime contains.

EXAMPLE 60:

Listen to Example 60

One of the problems with the augmented seconds is that - since there are two now - the listener is more likely to perceive them as minor thirds. That's pretty apparent in these super-progressions and super-regressions, where they are exposed in the bass.

*****

EXAMPLE 61:

Listen to Example 61

The Theta system is even stranger, as in addition to diminished thirds - which the listener is likely to perceive as major seconds - and the augmented seconds in the scale, we now have augmented thirds in the harmonies as well, which the listener will probably understand as perfect fourths. These distortions of native system realities can put listeners adrift, which is a nice way to affect them if that's the composer's desire. These systemic distortions of native reality are more compelling than the old techniques used in the serial systems of the century just passed, because a native system harmonic continuity can be progressively morphed into these alternate harmonic realities, and still retain their recognizability, as we will see in chapter twelve. One of the main criticisms of the various atonal methods is the, "any note could be replaced by any other note" feeling that listeners get, which is not an issue at all with systemic modifications from native, to exotic, and finally alien diatonic systems.

EXAMPLE 62:

Listen to Example 62

The funky thirds are quite apparent with the third movements in the bass with these half-progressions and half-regressions, but the connection to modal reality, though tenuous, is never entirely broken. This is far more effective than simply emerging listeners into total chaos, which they usually object to (And rightly so, in most instances). The exceptions, as always, involve situations such as film and stage vehicles, where there are extra-musical contextual defining elements at work. But for absolute music, especially, the alternate contextual morphologies available with exotic and alien systems are a great way to evoke the uncanny while still anchoring the listener to a modal locus.

EXAMPLE 63:

Listen to Example 63

The bizarre nature of Theta Prime is nicely exposed by the super-progressions and super-regressions.

*****

EXAMPLE 64:

Listen to Example 64

The minor tonic of Iota Prime adds even more darkness to the character of this alien system.

EXAMPLE 65:

Listen to Example 65

By the way, if you have been listening to all of the examples as we have progressed through the nine diatonic contextual systems, your brain is being hacked by them: Since almost nobody other than myself has ever listened to them systematically like this, I can tell you that this exposure will alter your conception of what musical reality can be. This is all leading up to the next chapter, where we will take an harmonic Mobius loop and morph it through all nine systems. If you have a musically sensitive mind, this will be very enlightening for you.

EXAMPLE 66:

Listen to Example 66

And so, all nine of the diatonic contextual systems have now been presented: Alpha, Beta, and Gamma for the three native systems; Delta, Epsilon, and Zeta for the three exotic systems; and finally Eta, Theta, and Iota for the alien systems. The total of sixty-three modal contexts and sub-contexts provide many more sonic resources than the old common practice guys were ever aware of, and even more than more modern jazz composers ever conceived of, all in a simple and intuitive system that can be applied to any diatonic theme a composer comes up with.

There are systems outside of the diatonic realm of course - which I have briefly alluded to previously - and we will begin to look at those in chapter thirteen.

I like that photo, even though she bears an uncanny resemblance to my ex-wife.

Why Music Works: Chapter Ten

2010-10-05T18:27:00.002-05:00

The Alien Diatonic Contextual Systems: Eta, Theta, and Iota

PREFACE to All Posts:

This is to be the culmination of the Musical Relativity series of posts I did back in 2006, which can be found to your right in the sidebar. Back then I was calling the series Musical Implications of the Harmonic Overtone Series. Even before that, I did a series of posts called Harmonic Implications of the Overtone Series that started this all. Here, I am presenting the final weblog version of the evolving book I've decided to publish with the intention of getting some feedback before I create the final print version, which I plan to put into the ePub format for iBooks. So, please feel free to ask any questions about anything that you think I haven't made perfectly clear, and don't hesitate to offer any constructive criticisms or suggestions. Since this project is the accidental result of several decades of curios inquiry - and many prominent and also relatively anonymous theorists and teachers have contributed ideas to it (Which I will credit where memory serves and honor dictates) - I am eager to get a final layer of polish from any and all who may happen to read this series and find it useful, or potentially so. Since I am creating these posts as .txt files first, revision should be a simple process.

Since my pre-degree studies at The Guitar Institute of the Southwest and my undergraduate work at Berklee College of Music looked at music theory from the jazz perspective, and then my master of music and doctor of musical arts studies at Texas State and The University of North Texas were from the traditional perspective, a large part of how I discovered the things in this book-in-progress was the result of my trying to reconcile those different theoretical viewpoints. Since I want this work to be of practical value, I have retained all of the traditional theoretical nomenclature possible, and only added to it where necessary to describe phenomena that have not heretofore been present in musical analysis. I have, however, standardized terminology into what I think is the most logical system yet devised, and that will be explained as the reader goes along. There is a lot of built-in review and repetition - something I've learned from my decades of private teaching - so even a once-through with this systematic approach to understanding musical phenomena ought to be of significant benefit.

Outright addition to traditional musical analysis is limited to the symbology required to label root motion and transformation types so that the root motion and transformation patterns are visible: This greatly facilitates comprehension, and since good and bad harmonic continuities are separated by the effectiveness or lack thereof in the root motion and transformation patterns, this also actually functions as an aid to composition. All symbology - old and new - has been worked out over the past three decades so that everything is readily available with the standard letters, numbers, and symbols found on a QWERTY keyboard.

Finally, for the contextual systems, I have used the Greek alphabet: The normal diatonic systems - those comprised of two minor seconds and five major seconds - are Alpha, Beta, and Gamma. The exotic diatonic systems - those that contain a single augmented second - are Delta, Epsilon, and Zeta, and finally, the alien diatonic systems - those that contain two augmented seconds - are Eta, Theta, and Iota. Since the theoretical writings that started western art music out were handed down from ancient Greece, I thought this would be a fitting tribute, as well as a handy and logical classification scheme.

Basically, if you have a baccalaureate-level understanding of music theory from either a jazz or traditional perspective, you should have no problem understanding anything in this straight-forward treatise.

INTRODUCTION to Chapter Ten:

In chapter one, I demonstrated how the overtone sonority generates the three normal diatonic systems - those seven note systems that contain two semitones and five tones: Alpha, Beta, and Gamma - and then in chapter two we examined each of those systems in detail, discovering that the primacy of Alpha is due to the fact that all seven of its harmonies can be arranged in progressive order. In chapter three, we examined the contextualization of Alpha Prime, looking at the various different root progressions types it can exhibit, their various transformations, and through this we also started to look into the world of musical effect and affect. Chapter four was dedicated to examining how Beta Prime and Gamma Prime compared to Alpha, using the same musical proof formats developed in chapter three. Through those proofs, we discovered some very unusual harmonic effects that evoke the uncanny that are contained in the Beta and Gamma systems. Chapter five then took us out of the diatonic harmonic world and into the chromatic realm as we discovered the origins of the secondary dominant sub-system sonorities. After the secondary dominants, in chapter six, we looked at the secondary subdominant sub-system of harmonies, which completed a larger set of integrated chromatic systems, which we will look at in detail later.

Then in chapter seven, we looked at the exotic diatonic systems - those seven note contextual systems that contain a single augmented second: Delta, Epsilon, and Zeta - and in chapter eight we looked in detail at the root motion types they contain, and the unique harmonic effects that these unusual systems create. Last time, in chapter nine, I demonstrated a phenomenon that is an artifact of patterned root progressions, which I pointed out earlier, and that is harmonic canon. Depending upon how harmonic canons are developed and set up, I showed how they can also exhibit the phenomena I call Musical Escher Morphs and Harmonic Mobius Loops.

Chapters ten and eleven will be devoted to the alien diatonic contextual systems - which are those seven note systems that contain two augmented seconds: Eta, Theta, and Iota - and with these chapters we will complete all nine master diatonic contextual systems and the total of sixty-three independent and dependent diatonic modes. Today's chapter ten will look at the genesis and structure of the alien systems.

*****

CHAPTER TEN:

EXAMPLE 52:

Listen to Example 52

The Eta Prime master context is generated by a V(d5m7) altered overtone sonority resolving to a major seventh tonic chord, and then on to a minor/major seventh on the subdominant degree. This yields two augmented seconds: One between the minor second and the major third in the lower tetrachord, and the other between the minor sixth and the leading tone in the upper tetrachord. I've heard this scale referred to as double harmonic major, but there are several double harmonic major modes in the exotic systems, so - since the tonic is a major seventh and the fourth degree is perfect - it is more precise to call it Ionian minor second, minor sixth.

Eta 4 deserves a mention here, as I first encountered this mode as double harmonic minor while back at Berklee in the early 80's. Since then I've also heard it called an Arabian scale or the snake charmer scale. It's actually pretty cool.

EXAMPLE 53:

Listen to Example 53

Theta Prime is generated by a V(d5M7) on the dominant degree - and now you can see why these systems become alien: There is no primary tritone in some of the dominant stand-in sonorities - which resolves again to a major seventh tonic, but now the chord on the raised subdominant degree is the very strange #iv(d3d5d7) sonority. Since the fourth degree is now augmented, that makes this a Lydian minor second, minor sixth (Which is a different species of, "double harmonic major").

EXAMPLE 54:

Listen to Example 54

For Iota Prime, we are now resolving the V(d5M7) sonority into a minor/major seventh tonic, while the raised fourth degree still supports a #iv(d3d5d7) chord. As you can probably guess, there are going to be some very strange effects within these alien systems.

*****

EXAMPLE 55:

Listen to Example 55 (The example is the harmonized scale only).

Eta prime has the rare feature of being an intervalic palindrome, as it's intervals read the same forwards or backwards: 1, 3, 1, 2, 1, 3, 1. In the alpha system, this honor goes to Alpha 2, which is the Dorian mode. Eta 2, Eta 3, and Eta 4 are nominally independent since there is an harmony on the unaltered dominant degree and a functional tonic triad, however they are not simple to establish independently in practice, but it can be done. The other three sub-contexts are obviously dependent.

EXAMPLE 56:

Listen to Example 56 (The example is the harmonized scale only).

I neglected to put the intervals in example fifty-six, but they go; 1, 3, 2, 1, 1, 3, 1. There is only one independent sub-context in the Theta system - Theta 3 (Mistakenly labeled dependent) - because of a new phenomenon that destroys a tonic triad: an augmented third in the case of Theta 2 and a diminished third in the case of Theta 7. Obviously, this is a difficult system to work in, even with the independent modal sub-contexts.

EXAMPLE 57:

Listen to Example 57 (The example is the harmonized scale only).

There is a good case to be made for calling Iota 6 the prime here, as it has an actual altered dominant as well as a natural fourth degree. If you'll notice, though, I organized all of the diatonic systems around C-natural in such a way as to start with the fewest accidentals, and add from there. In this case, Eta Prime only required a D-flat and an A-flat, Theta Prime added the F-sharp to that, and Iota Prime here got the additional E-flat. Since I just recently worked these alien forms out, I may rethink my organization before the final version.

I again forgot to put the intervals under the harmonized scale, but it's; 1, 2, 3, 1, 1, 3, 1.

Next time we'll break out the musical proofs to look at and listen to the root progressions and transformations in the alien systems.

Why Music Works: Chapter Nine

2010-10-01T03:27:00.000-05:00

Harmonic Canons and Musical Escher Morphs

PREFACE to All Posts:

This is to be the culmination of the Musical Relativity series of posts I did back in 2006, which can be found to your right in the sidebar. Back then I was calling the series Musical Implications of the Harmonic Overtone Series. Even before that, I did a series of posts called Harmonic Implications of the Overtone Series that started this all. Here, I am presenting the final weblog version of the evolving book I've decided to publish with the intention of getting some feedback before I create the final print version, which I plan to put into the ePub format for iBooks. So, please feel free to ask any questions about anything that you think I haven't made perfectly clear, and don't hesitate to offer any constructive criticisms or suggestions. Since this project is the accidental result of several decades of curios inquiry - and many prominent and also relatively anonymous theorists and teachers have contributed ideas to it (Which I will credit where memory serves and honor dictates) - I am eager to get a final layer of polish from any and all who may happen to read this series and find it useful, or potentially so. Since I am creating these posts as .txt files first, revision should be a simple process.

Since my pre-degree studies at The Guitar Institute of the Southwest and my undergraduate work at Berklee College of Music looked at music theory from the jazz perspective, and then my master of music and doctor of musical arts studies at Texas State and The University of North Texas were from the traditional perspective, a large part of how I discovered the things in this book-in-progress was the result of my trying to reconcile those different theoretical viewpoints. Since I want this work to be of practical value, I have retained all of the traditional theoretical nomenclature possible, and only added to it where necessary to describe phenomena that have not heretofore been present in musical analysis. I have, however, standardized terminology into what I think is the most logical system yet devised, and that will be explained as the reader goes along. There is a lot of built-in review and repetition - something I've learned from my decades of private teaching - so even a once-through with this systematic approach to understanding musical phenomena ought to be of significant benefit.

Outright addition to traditional musical analysis is limited to the symbology required to label root motion and transformation types so that the root motion and transformation patterns are visible: This greatly facilitates comprehension, and since good and bad harmonic continuities are separated by the effectiveness or lack thereof in the root motion and transformation patterns, this also actually functions as an aid to composition. All symbology - old and new - has been worked out over the past three decades so that everything is readily available with the standard letters, numbers, and symbols found on a QWERTY keyboard.

Finally, for the contextual systems, I have used the Greek alphabet: The normal diatonic systems - those comprised of two minor seconds and five major seconds - are Alpha, Beta, and Gamma. The exotic diatonic systems - those that contain a single augmented second - are Delta, Epsilon, and Zeta, and finally, the alien diatonic systems - those that contain two augmented seconds - are Eta, Theta, and Iota. Since the theoretical writings that started western art music out were handed down from ancient Greece, I thought this would be a fitting tribute, as well as a handy and logical classification scheme.

Basically, if you have a baccalaureate-level understanding of music theory from either a jazz or traditional perspective, you should have no problem understanding anything in this straight-forward treatise.

INTRODUCTION to Chapter Eight:

In chapter one, I demonstrated how the overtone sonority generates the three normal diatonic systems - those seven note systems that contain two semitones and five tones: Alpha, Beta, and Gamma - and then in chapter two we examined each of those systems in detail, discovering that the primacy of Alpha is due to the fact that all seven of its harmonies can be arranged in progressive order. In chapter three, we examined the contextualization of Alpha Prime, looking at the various different root progressions types it can exhibit, their various transformations, and through this we also started to look into the world of musical effect and affect. Chapter four was dedicated to examining how Beta Prime and Gamma Prime compared to Alpha, using the same musical proof formats developed in chapter three. Through those proofs, we discovered some very unusual harmonic effects that evoke the uncanny that are contained in the Beta and Gamma systems. Chapter five then took us out of the diatonic harmonic world and into the chromatic realm as we discovered the origins of the secondary dominant sub-system sonorities. After the secondary dominants, in chapter six, we looked at the secondary subdominant sub-system of harmonies, which completed a larger set of integrated chromatic systems, which we will look at in detail later.

Then in chapter seven, we looked at the exotic diatonic systems - those seven note contextual systems that contain a single augmented second: Delta, Epsilon, and Zeta - and in chapter eight we looked in detail at the root motion types they contain, and the unique harmonic effects that these unusual systems create.

Now, in chapter nine, I will demonstrate a phenomenon that is an artifact of patterned root progressions, which I pointed out earlier, and that is harmonic canon. Depending upon how harmonic canons are developed and set up, they can also exhibit the phenomena I call Musical Escher Morphs and Harmonic Mobius Loops.

*****

CHAPTER NINE:

EXAMPLE 49:

Listen to Example 49

On the top system we have the end-contextualized diatonic direct transforming progressive root motion example that we first saw way back in example seven when we initially arranged the harmonies of Alpha Prime in progressive order. As I pointed out when we contextualized that continuity for example ten, an artifact of the constant progressive root motions is an harmonic canon; specifically, a double canon at the fourth above. This means that the harmonic series, progressing most naturally, produces canon: It is an entirely natural phenomenon.

The second system shows the extracted canon, which is still diatonic, and so it doesn't draw much attention to itself. If, however, we begin to embellish the diatonic version with secondary dominants and make all of the target chords minor, we get the strict canon on the third system. Penultimately, we can further adorn the canon with secondary V(d5m7) sonorities, as we have on the fourth system. Now it's very obviously a double canon. Finally, if we dovetail all of these versions together - diatonic, secondary dominant, and secondary V(d5m7) - we end up with the Musical Escher Morph on the fifth system.

I call these Musical Escher Morphs for reasons that should be obvious: They are a musical analog to this:

One harmonic form transforms into another over successive modulations of the root motion pattern. Realize that setting up this pure harmony version of the Musical Escher Morph is just the first step on the path to creating a final composition. Through further elaboration - which would take us into the realm of melody, and so is beyond the scope of this section of WMW - we could end up with something akin to Pachelbel's Canon in D, but much more modern and technologically proficient.

This repeating single-interval root motion is just the most basic kind of succession that creates harmonic canon as an artifact, however. Repeating root motion patterns of two intervals - like the one in Pachelbel's canon - also produce harmonic canons as artifacts.

*****

EXAMPLE 50:

Listen to Example 50

Here, on the top system, I have constructed a direct transforming harmonic continuity that consists of an half-progression alternating with a progression through diatonic Alpha Prime. The extracted diatonic canon on the second system reveals it as a four-part canon at the second above - not a double canon as before - but the continuity actually ends before the full canon is complete. When you have more than one root motion type, the transformations can allow each voice to play every part in the harmonies - root, third, fifth, and seventh - and so true four-voice canons can result.

The incompleteness of the diatonic canon coupled with the two intervals - descending minor third and ascending perfect fourth - presents me with the opportunity to create a two-interval twelve tone row for the bass part, and that creates the Musical Escher Morph on systems four and five (Sorry for the double bar line in the middle of that; just noticed). Since I introduced first secondary dominants and then secondary V(d5m7) chords in that one, there are interrupted crosswise transformations now, and so the four-part canon is at the unison. Again, this is just the skeleton of what the final canon could become through melodic elaboration, and yes, I plan to use this in a larger composition at some point. It's really quite wonderful.

*****

EXAMPLE 51:

Listen to Example 51

Justly, the most famous of all harmonic canons is Pachelbel's Canon in D, and the original continuity that Johann wrote is on the top system. This is a continuity of two root motion types as well, it being a regression followed by a super-progression. However, in the diatonic version the first super-progression is up by whole step, and the second is by half step. Now, Pachelbel composed this in the years just before J.S. Bach was born, so he didn't know anything about pure harmony, but it does tell us that the intuition of composers had figured out that repeating patterns in the bass could support canons as far back as three-hundred-twenty-five years ago. That's pretty amazing.

So, on the second system, I have converted Pachelbel's continuity into modern pure transformational harmony. This reveals to us that the underlying canon is at the sixth above - or third below - as we see on system three. If you are familiar with Pachelbel's version - and who isn't - you'll remember lots of parallel thirds and sixths, so at some level he figured this out too.

On the bottom two systems I have extended Pachelbel's continuity by making the original root progressions strict: Regression followed by super-progression of a whole step (Again, sorry for the double bar line there; I'll have to fix that for the final examples). This does not create a twelve tone row, as there are only eight pitch classes in the bass line, but it does create a direct modulation a tritone away - way gnarly - and that creates a very special kind of Musical Escher Morph that is also an Harmonic Mobius Loop.

Escher himself did a famous Mobius strip with ants.

I think I first saw that when I was about ten. Quite fascinating.

In an Harmonic Mobius Loop the root motion types are equalized so that musical gravity and anti-gravity are balanced out, and the end of the transformation runs back into the beginning. As you can see by comparing the first measure on the fourth system with the first measure on the fifth system - both over a I(M7) sonority - that is the case here. I break the strictness of the root motion pattern to bring the piece to an end, but I could just as easily repeat the first eight measures ad infinitum without the transformational stratum moving up or down at all. Yes, that's the plan for the final composition this will be in, as I plan to use this too.

We will look at Harmonic Mobius Loops more in chapter eleven, but chapter ten will present the three alien diatonic contextual systems of Eta, Theta, and Iota.

*****

Now that is an awesome redhead.

Why Music Works: Chapter Eight

2010-09-28T15:27:00.000-05:00

Root Motion and Transformation Types in the Delta, Epsilon, and Zeta Systems

PREFACE to All Posts:

This is to be the culmination of the Musical Relativity series of posts I did back in 2006, which can be found to your right in the sidebar. Back then I was calling the series Musical Implications of the Harmonic Overtone Series. Even before that, I did a series of posts called Harmonic Implications of the Overtone Series that started this all. Here, I am presenting the final weblog version of the evolving book I've decided to publish with the intention of getting some feedback before I create the final print version, which I plan to put into the ePub format for iBooks. So, please feel free to ask any questions about anything that you think I haven't made perfectly clear, and don't hesitate to offer any constructive criticisms or suggestions. Since this project is the accidental result of several decades of curios inquiry - and many prominent and also relatively anonymous theorists and teachers have contributed ideas to it (Which I will credit where memory serves and honor dictates) - I am eager to get a final layer of polish from any and all who may happen to read this series and find it useful, or potentially so. Since I am creating these posts as .txt files first, revision should be a simple process.

Since my pre-degree studies at The Guitar Institute of the Southwest and my undergraduate work at Berklee College of Music looked at music theory from the jazz perspective, and then my master of music and doctor of musical arts studies at Texas State and The University of North Texas were from the traditional perspective, a large part of how I discovered the things in this book-in-progress was the result of my trying to reconcile those different theoretical viewpoints. Since I want this work to be of practical value, I have retained all of the traditional theoretical nomenclature possible, and only added to it where necessary to describe phenomena that have not heretofore been present in musical analysis. I have, however, standardized terminology into what I think is the most logical system yet devised, and that will be explained as the reader goes along. There is a lot of built-in review and repetition - something I've learned from my decades of private teaching - so even a once-through with this systematic approach to understanding musical phenomena ought to be of significant benefit.

Outright addition to traditional musical analysis is limited to the symbology required to label root motion and transformation types so that the root motion and transformation patterns are visible: This greatly facilitates comprehension, and since good and bad harmonic continuities are separated by the effectiveness or lack thereof in the root motion and transformation patterns, this also actually functions as an aid to composition. All symbology - old and new - has been worked out over the past three decades so that everything is readily available with the standard letters, numbers, and symbols found on a QWERTY keyboard.

Finally, for the contextual systems, I have used the Greek alphabet: The normal diatonic systems - those comprised of two minor seconds and five major seconds - are Alpha, Beta, and Gamma. The exotic diatonic systems - those that contain a single augmented second - are Delta, Epsilon, and Zeta, and finally, the alien diatonic systems - those that contain two augmented seconds - are Eta, Theta, and Iota. Since the theoretical writings that started western art music out were handed down from ancient Greece, I thought this would be a fitting tribute, as well as a handy and logical classification scheme.

Basically, if you have a baccalaureate-level understanding of music theory from either a jazz or traditional perspective, you should have no problem understanding anything in this straight-forward treatise.

INTRODUCTION to Chapter Eight:

In chapter one, I demonstrated how the overtone sonority generates the three normal diatonic systems - those seven note systems that contain two semitones and five tones: Alpha, Beta, and Gamma - and then in chapter two we examined each of those systems in detail, discovering that the primacy of Alpha is due to the fact that all seven of its harmonies can be arranged in progressive order. In chapter three, we examined the contextualization of Alpha Prime, looking at the various different root progressions types it can exhibit, their various transformations, and through this we also started to look into the world of musical effect and affect. Chapter four was dedicated to examining how Beta Prime and Gamma Prime compared to Alpha, using the same musical proof formats developed in chapter three. Through those proofs, we discovered some very unusual harmonic effects that evoke the uncanny that are contained in the Beta and Gamma systems. Chapter five then took us out of the diatonic harmonic world and into the chromatic realm as we discovered the origins of the secondary dominant sub-system sonorities. Previously, in chapter six, we looked at the secondary subdominant sub-system of harmonies, which completed a larger set of integrated chromatic systems, which we will look at in detail later.

Last time, in chapter seven, we went back a bit and looked at the exotic diatonic systems - those seven note contextual systems that contain a single augmented second: Delta, Epsilon, and Zeta - and now in chapter eight we'll look in detail at the root motion types they contain, and the unique harmonic effects that these unusual systems create.

*****

CHAPTER EIGHT:

EXAMPLE 40:

Listen to Example 40

Here we have the progressions and regressions in Delta Prime, using the same end-contextualized musical proofs I've presented earlier for Alpha, Beta and Gamma. One nice thing about using musical proofs that the reader can listen to, is that I really don't have to explain very much now that all of the elements of the contextual system concept have been previously presented. All of these transformations are direct, as the point now is to hear the various harmonies in isolation, so as to hear their uniqueness. The bVI(A5M7) is a hot sonority, and being entered via quadra-tone and exited by tritone - or the other way around in the case of the regressions - really puts it in stark relief here.

EXAMPLE 41:

Listen to Example 41

It really is difficult to even notice the augmented second movements in the transformational stratum unless you are careful to listen for them.

EXAMPLE 42:

Listen to Example 42

As I pointed out above, unusual harmonies more or less alternate with more normal seventh chords in this example. That's a nice resource. Now, on to the Epsilon system.

*****

EXAMPLE 43:

Listen to Example 43

This is the system normally referred to as harmonic minor, so some of these effects may be familiar to you. Since this should all be old hat now, I'll firmally dispense with the observations, unless something truly unique arises.

EXAMPLE 44:

Listen to Example 44

EXAMPLE 45:

Listen to Example 45

Now, on to the Zeta system, which is like melodic minor with a Phrygian minor second.

*****

EXAMPLE 46:

Listen to Example 46

NOTE: The D in the penultimate measure of the top system appears as a D-natural instead of the D-flat it ought to be. I corrected this when I was proofing the audio examples, but I'd already uploaded the JPEG files by then. So, it looks wrong, but it sounds right. A big part of this post series is to work the kinks out of the examples and more perfectly define the presentation order (More on that in a few minutes).

Note also that in the progressive root motion from the bIII(A5m7) - normally called an augmented seventh chord - that there is the augmented second in the transformation as the B-natural moves down to A-flat: This is why the so-called augmented seventh chords do not fit into the secondary dominant galaxy of sonorities - the augmented fifth has to move an augmented second down to get to the new root.

There are some very wicked sounding sonorities in this system. The vii(d3d5d7) is particularly cool.

EXAMPLE 47

Listen to Example 47

EXAMPLE 48

Listen to Example 48

Lots more interesting sonic resources in the exotic systems, but wait until we get to the alien systems (Those with two augmented seconds). I worked those out today - I could swear I did that before - and they are really, really creepy. One of the sub-contexts is the so-called Arabian or snake charmer scale, and that system creates some very bizarre sonorities. Next time, however, we are going to look at harmonic canons.

I think I have the final chapter outline done for the book now. Instead of presenting the secondary dominant sub-system and the secondary subdominant sub-system together, as I've done in this series, I'm going to break them up like so.

01] The Harmonic Series: Its Structure, Forces, and Primordial Resolution
02] Genesis of the Native Diatonic Contextual Systems: Alpha, Beta, and Gamma
03] Root Motion and Voice Transformation in the Native Diatonic Contextual Systems
04] Sonorities of the Secondary Dominant Contextual Sub-System
05] Genesis of the Exotic Diatonic Contextual Systems: Delta, Epsilon, and Zeta
06] Root Motion and Voice Transformation in the Exotic Diatonic Contextual Systems
07] Sonorities of the Secondary Subdominant Contextual Sub-System
08] Genesis of the Alien Diatonic Contextual Systems: Eta, Theta, and Iota
09] Root Motion and Voice Transformation in the Alien Diatonic Contextual Systems
10] Harmonic Canons, Musical Escher Morphs, and Musical Mobius Loops
11] Genesis of the Hybrid Nonatonic Contextual Systems: Kappa, Lambda, and Mu
12] The Integrated Chromatic Contextual Systems: Chi, Psi, and Omega

One thing I wanted to avoid, was putting all of the contextual systems together at the beginning. Not only can it get tedious that way, but spacing them out lends itself to the built-in review device that I like to use when teaching.

*****

Why Music Works: Chapter Seven

2010-09-26T06:27:00.001-05:00

The Exotic Diatonic Contextual Systems: Delta, Epsilon, and Zeta

PREFACE to All Posts:

This is to be the culmination of the Musical Relativity series of posts I did back in 2006, which can be found to your right in the sidebar. Back then I was calling the series Musical Implications of the Harmonic Overtone Series. Even before that, I did a series of posts called Harmonic Implications of the Overtone Series that started this all. Here, I am presenting the final weblog version of the evolving book I've decided to publish with the intention of getting some feedback before I create the final print version, which I plan to put into the ePub format for iBooks. So, please feel free to ask any questions about anything that you think I haven't made perfectly clear, and don't hesitate to offer any constructive criticisms or suggestions. Since this project is the accidental result of several decades of curios inquiry - and many prominent and also relatively anonymous theorists and teachers have contributed ideas to it (Which I will credit where memory serves and honor dictates) - I am eager to get a final layer of polish from any and all who may happen to read this series and find it useful, or potentially so. Since I am creating these posts as .txt files first, revision should be a simple process.

Since my pre-degree studies at The Guitar Institute of the Southwest and my undergraduate work at Berklee College of Music looked at music theory from the jazz perspective, and then my master of music and doctor of musical arts studies at Texas State and The University of North Texas were from the traditional perspective, a large part of how I discovered the things in this book-in-progress was the result of my trying to reconcile those different theoretical viewpoints. Since I want this work to be of practical value, I have retained all of the traditional theoretical nomenclature possible, and only added to it where necessary to describe phenomena that have not heretofore been present in musical analysis. I have, however, standardized terminology into what I think is the most logical system yet devised, and that will be explained as the reader goes along. There is a lot of built-in review and repetition - something I've learned from my decades of private teaching - so even a once-through with this systematic approach to understanding musical phenomena ought to be of significant benefit.

Outright addition to traditional musical analysis is limited to the symbology required to label root motion and transformation types so that the root motion and transformation patterns are visible: This greatly facilitates comprehension, and since good and bad harmonic continuities are separated by the effectiveness or lack thereof in the root motion and transformation patterns, this also actually functions as an aid to composition. All symbology - old and new - has been worked out over the past three decades so that everything is readily available with the standard letters, numbers, and symbols found on a QWERTY keyboard.

Finally, for the contextual systems, I have used the Greek alphabet: The normal diatonic systems - those comprised of two minor seconds and five major seconds - are Alpha, Beta, and Gamma. The exotic diatonic systems - those that contain a single augmented second - are Delta, Epsilon, and Zeta, and finally, the alien diatonic systems - those that contain two augmented seconds - are Eta, Theta, and Iota. Since the theoretical writings that started western art music out were handed down from ancient Greece, I thought this would be a fitting tribute, as well as a handy and logical classification scheme.

Basically, if you have a baccalaureate-level understanding of music theory from either a jazz or traditional perspective, you should have no problem understanding anything in this straight-forward treatise.

INTRODUCTION to Chapter Seven:

In chapter one, I demonstrated how the overtone sonority generates the three normal diatonic systems - those seven note systems that contain two semitones and five tones: Alpha, Beta, and Gamma - and then in chapter two we examined each of those systems in detail, discovering that the primacy of Alpha is due to the fact that all seven of its harmonies can be arranged in progressive order. In chapter three, we examined the contextualization of Alpha Prime, looking at the various different root progressions types it can exhibit, their various transformations, and through this we also started to look into the world of musical effect and affect. Chapter four was dedicated to examining how Beta Prime and Gamma Prime compared to Alpha, using the same musical proof formats developed in chapter three. Through those proofs, we discovered some very unusual harmonic effects that evoke the uncanny that are contained in the Beta and Gamma systems. Chapter five then took us out of the diatonic harmonic world and into the chromatic realm as we discovered the origins of the secondary dominant sub-system sonorities. Previously, in chapter six, we looked at the secondary subdominant sub-system of harmonies, which completed a larger set of integrated chromatic systems, which we will look at in detail later.

At this point, in chapter seven, we will go back a bit, in a way, by looking at the exotic diatonic systems - those seven note contextual systems that contain a single augmented second: Delta, Epsilon, and Zeta - and then in chapter eight we'll look in detail at the root motion types they contain, and the unique harmonic effects that these unusual systems create.

*****

CHAPTER SEVEN:

The Delta diatonic contextual system is created by an overtone sonority resolving to a major tonic, and then to a minor subdominant. Since the rules for diatonic resolutional genesis call for retaining the inflection of notes that appear in previous harmonies, that means that the subdominant minor triad carries a hotly dissonant major seventh (m3, P5, and M7 create an augmented triad in this chord).

EXAMPLE 34:

Since resolutional genesis is now a familiar concept, audio examples will be saved for the progressive orientation examples from here on out.

OBSERVATIONS:

1. Augmented seconds are perfectly acceptable in harmonic transformations.

2. A minor, major-seventh chord makes a perfectly acceptable subdominant harmony.

3. Delta Prime is often called, "harmonic major" but Ionian minor sixth is more descriptively accurate.

4. The Delta Prime tonic scale is, 2, 2, 1, 2, 1, 3, 1: Two whole steps are adjacent at the beginning of the mode.

*****

The Epsilon diatonic contextual system is created by an overtone sonority resolving to a minor tonic, and then onto a minor subdominant.

EXAMPLE 35:

OBSERVATIONS:

1. Epsilon Prime is often called, "harmonic minor" but Aeolian major seventh is more descriptively accurate.

2. The Epsilon Prime tonic scale is, 2, 1, 2, 2, 1, 3, 1: Two whole steps are adjacent in the middle of the mode.

*****

The Zeta diatonic contextual system is created by an overtone chord with a diminished fifth resolving to a minor tonic, and then into a minor subdominant.

EXAMPLE 36:

OBSERVATIONS:

1. Zeta prime is often called, "Phrygian harmonic" but Phrygian major seventh is more descriptively accurate.

2. The Zeta Prime scale is, 1, 2, 2, 2, 1, 3, 1: The three whole steps are adjacent.

*****

Here are the displacement modes of the Delta contextual system.

EXAMPLE 37A:

Four harmonies of the Delta system can be arranged in progressive order.

EXAMPLE 37B:

Listen to Example 37B

As you can hear, the augmented second in the final transformation does not sound overly strange.

OBSERVATIONS:

1. Delta or Epsilon can be considered as a point of origin for the V(m7m9) - vii(d5d7) sonorities.

Plainly, these harmonies are an artifact created through the genesis of these systems.

2. As with Beta, the Delta system has three independent and three dependent sub-contexts.

3. A m(d5d7) sonority can function perfectly well as a dominant with progressive root motion.

That would be as in Delta 3 above.

*****

Here are the displacement modes of the Epsilon contextual system.

EXAMPLE 38A:

Four harmonies of the Epsilon system can be arranged in progressive order.

EXAMPLE 38B:

Listen to Example 38B

If you were into 80's and 90's virtuoso rock guitar, you might even detect a little of the feel of the background for some of Yngwie Malmsteen's stuff in that, as I do. He was very fond of Epsilon Prime aka "harmonic minor."

OBSERVATIONS:

1. Like Beta and Delta, Epsilon has three independent and three dependent sub-contexts.

*****

Here are the displacement modes of the Zeta contextual system.

EXAMPLE 39A:

Only three harmonies of the Zeta system can be arranged in progressive order.

EXAMPLE 39B:

Listen to Example 39B

OBSERVATIONS:

1. Delta, Epsilon, and Zeta each have three independent and three dependent sub-contexts.

2. The Delta, Epsilon, and Zeta systems add an additional 21 modes to the 21 that resulted from the genesis of the Alpha, Beta and Gamma systems.

So, we're up to a total of 42 diatonic modes now: 21 normal modes and 21 exotic modes.

3. Three additional contextual systems are available with two augmented seconds.

As I mentioned previously, I thought I had worked these out, but I can't locate them now, so I'm not positive; there may be only two more. In any event, there will be a brief intermission as I create some more examples, because this is the end of the examples I created back in 2008 before I moved from Alpine to San Antonio. When we do continue, it will be to look at and listen to the various root motion and transformation types that Delta, Epsilon, and Zeta exhibit, using the same proofs we used for Alpha, Beta, and Gamma.

Since this post will put chapter one off of the home page, there is now a section in the sidebar for this series.

It's all drama with some girls.

Why Music Works: Chapter Six

2010-09-22T02:27:00.003-05:00

The Harmonies of the Secondary Subdominant Contextual Sub-System

PREFACE to All Posts:

This is to be the culmination of the Musical Relativity series of posts I did back in 2006, which can be found to your right in the sidebar. Back then I was calling the series Musical Implications of the Harmonic Overtone Series. Even before that, I did a series of posts called Harmonic Implications of the Overtone Series that started this all. Here, I am presenting the final weblog version of the evolving book I've decided to publish with the intention of getting some feedback before I create the final print version, which I plan to put into the ePub format for iBooks. So, please feel free to ask any questions about anything that you think I haven't made perfectly clear, and don't hesitate to offer any constructive criticisms or suggestions. Since this project is the accidental result of several decades of curios inquiry - and many prominent and also relatively anonymous theorists and teachers have contributed ideas to it (Which I will credit where memory serves and honor dictates) - I am eager to get a final layer of polish from any and all who may happen to read this series and find it useful, or potentially so. Since I am creating these posts as .txt files first, revision should be a simple process.

Since my pre-degree studies at The Guitar Institute of the Southwest and my undergraduate work at Berklee College of Music looked at music theory from the jazz perspective, and then my master of music and doctor of musical arts studies at Texas State and The University of North Texas were from the traditional perspective, a large part of how I discovered the things in this book-in-progress was the result of my trying to reconcile those different theoretical viewpoints. Since I want this work to be of practical value, I have retained all of the traditional theoretical nomenclature possible, and only added to it where necessary to describe phenomena that have not heretofore been present in musical analysis. I have, however, standardized terminology into what I think is the most logical system yet devised, and that will be explained as the reader goes along. There is a lot of built-in review and repetition - something I've learned from my decades of private teaching - so even a once-through with this systematic approach to understanding musical phenomena ought to be of significant benefit.

Outright addition to traditional musical analysis is limited to the symbology required to label root motion and transformation types so that the root motion and transformation patterns are visible: This greatly facilitates comprehension, and since good and bad harmonic continuities are separated by the effectiveness or lack thereof in the root motion and transformation patterns, this also actually functions as an aid to composition. All symbology - old and new - has been worked out over the past three decades so that everything is readily available with the standard letters, numbers, and symbols found on a QWERTY keyboard.

Finally, for the contextual systems, I have used the Greek alphabet: The normal diatonic systems - those comprised of two minor seconds and five major seconds - are Alpha, Beta, and Gamma. The exotic diatonic systems - those that contain a single augmented second - are Delta, Epsilon, and Zeta, and finally, the alien diatonic systems - those that contain two augmented seconds - are Eta, Theta, and Iota. Since the theoretical writings that started western art music out were handed down from ancient Greece, I thought this would be a fitting tribute, as well as a handy and logical classification scheme.

Basically, if you have a baccalaureate-level understanding of music theory from either a jazz or traditional perspective, you should have no problem understanding anything in this straight-forward treatise.

INTRODUCTION to Chapter Six:

In chapter one, I demonstrated how the overtone sonority generates the three normal diatonic systems - those seven note systems that contain two semitones and five tones: Alpha, Beta, and Gamma - and then in chapter two we examined each of those systems in detail, discovering that the primacy of Alpha is due to the fact that all seven of its harmonies can be arranged in progressive order. In chapter three, we examined the contextualization of Alpha Prime, looking at the various different root progressions types it can exhibit, their various transformations, and through this we also started to look into the world of musical effect and affect. Chapter four was dedicated to examining how Beta Prime and Gamma Prime compared to Alpha, using the same musical proof formats developed in chapter three. Through those proofs, we discovered some very unusual harmonic effects that evoke the uncanny that are contained in the Beta and Gamma systems. Chapter five then took us out of the diatonic harmonic world and into the chromatic realm as we discovered the origins of the secondary dominant sub-system sonorities.

Now in chapter six, we will look at the secondary subdominant sub-system of harmonies. Whereas in the secondary dominant sub-system the harmonies were all overtone chords or altered overtone chords, in the secondary subdominant sub-system the harmonies are actually different genders of chords: Major sevenths, minor sevenths, and dominant sevenths.

*****

CHAPTER SIX:

Secondary dominants are more well known than secondary subdominants, but both contextual sub-systems offer sonic resources that the composer ought to be aware of. The intuition of many rock music writers has lead them to the secondary subdominant major triads over the years - most notably to me, Pete Townshend of The Who - and this is partly explained by the fact that major triads sound so good with overdriven guitar amps. But, secondary subdominant major triads also produce a unique sonic environment that can't be duplicated in any other way. For example, The Real Me from The Who's Quadrophenia album is just loaded with them - as are many of the songs in that rock opera concept - and this is exactly where I started to figure them out when I was back in high school: I was positively addicted to those records.

Traditionally, the secondary subdominant major triads and major sevenths have been justified through the concept of modal interchange, which is borrowing harmonies from modes parallel to whichever one you have nominated as home. While this is a very useful compositional concept - I use it all the time - in this chapter we will see how the origin of these harmonies is better explained by the progressive resolutional paradigm established by the overtone chord.

In the modal interchange model, if we are in Alpha Prime - traditional major, or Ionian - the idea is to borrow the subdominant Lydian chords from the other Alpha System parallel modes. With I(M7) as the Ionian tonic, we get the bII(M7) Lydian chord from the parallel Phrygian, the bIII(M7) from the parallel Dorian, the IV(M7) is the native Ionian subdominant, bV(M7) comes from Locrian, the bVI(M7) comes from Aeolian, and finally, the bVII(M7) comes from Mixolydian. This creates a hybrid system of all subdominant chords surrounding a tonic much like the secondary dominants create a hybrid system of all dominants surrounding a tonic (And you can just as easily justify secondary dominants through modal interchange too). Later, when we look at integrated chromatic contextual systems, we will find that the secondary subdominants progressively leading away from the tonic create a grand subdominant preparation for the most remote of the secondary dominants, which then lead back to the tonic. Taking musical gravity into account, this creates a descending barber pole loop that spirals ever downward.

For now, we will simply look at the different genders of secondary subdominant, and where they came from.

EXAMPLE 28:

Listen to Example 28

As you can now see, the secondary subdominants are also generated by the resolutional paradigm established by the primordial resolution of the overtone sonority. Previously, with the secondary dominants, we were seeing overtone chords and altered overtone chords on the diatonic degrees progressively moving toward the tonic, whereas here, we see Lydian harmonies progressively leading away from the tonic, and so into the chromatic realm, at bVII(M7) and on. Where the connection between the secondary dominant sub-system and the secondary subdominant sub-system occurs is at the enharmonic progressive root motion that would be from the bV(M7) - the most remote of the secondary subdominants - to the V(d5m7)/iii - the most remote of the secondary dominants. Since G-flat is the enharmonic of F-sharp, this would lead to B-natural in the Alpha Prime on C here. As I mentioned above, we will look at this when we get to integrated chromatic contextual systems.

OBSERVATIONS:

1. The secondary subdominant major seventh chords are Lydian sonorities extending progressively away from the tonic.

2. This is opposed to the secondary dominant Mixolydian sonorities and altered Mixolydian sonorities, that approach the tonic progressively.

3. Upper structure tones for secondary subdominant major sevenths are: Major ninth, augmented eleventh, and major thirteenth.

4. As the iv(m7) chord in the penultimate measure shows, secondary subdominants can also me minor seventh chords.

Obviously, the minor subdominant was a better choice here because of all of the flatted notes in the bV(M7), which would have produced two augmented seconds going into a major chord on the fourth degree with the normal clockwise transformation that a super-regression carries.

5. Secondary subdominant major sevenths are generated by Alpha Prime (Ionian/Pure Major).

6. So, secondary subdominant minor sevenths will be generated by Alpha 6 (Aeolian/Pure Minor).

7. Therefore, secondary subdominant minor sevenths will be Dorian sonorities.

*****

EXAMPLE 29:

Listen to Example 29

This is example twenty-eight in the Alpha 6 independent sub-context.

OBSERVATIONS:

1. Secondary subdominant minor seventh chords are Dorian sonorities extending progressively away from the minor tonic.

2. This is opposed to the secondary dominant Mixolydian sonorities, which approach the tonic progressively.

3. Upper structure tones for secondary subdominant minor sevenths are: Major ninth, perfect eleventh, and major thirteenth.

4. The direct half-step resolution of the enharmonic V(d5m7m9)/V justifies the jazz subV7 practice.

Just as I was providentially able to present a minor subdominant in example twenty-eight, I was able to work in a V(d5m7m9)/V harmony here (The point of origin for the German Augmented Sixth in paleo-terminology). That sonority is enharmonic because of the tied G-flat, which would otherwise be an F-sharp. The resulting structure in the transformational stratum therefore reads like an A-flat dominant seventh chord, which jazz theorists call a subV7/V (Which shares its tritone with the dominant on D-natural), and all four notes move down by semitone into the primary dominant in this direct transformation. So, a jazz substitute secondary dominant resolving down by semitone in parallel is actually a V(d5m7m9/0) notated enharmonically and transforming directly. Pretty funny.

5. However, that direct resolution of the enharmonic V(d5m7m9)/V also nullifies jazz subV7 theory.

6. Additionally, that resolution of the V(d5m7m9)/V also nullifies traditional "German Sixth" theory.

If you could even say that the traditional German Augmented Sixth nomenclature rises to the level of a theory.

7. Both traditional and jazz theories are wrong here, but at least the traditional notation is correct.

8. Secondary subdominants can also be overtone sonorities generated by Beta Prime.

This is, in a way, out of bounds for the pure secondary subdominants generated by Alpha Prime and Alpha 6, but since Alpha 6 is combined with Beta Prime to create the traditionally so-called melodic minor nonatonic hybrid contextual system, I thought I'd go ahead and present them.

*****

EXAMPLE 30:

Listen to Example 30

This is example twenty-eight with secondary subdominant overtone chords.

OBSERVATIONS:

1. Secondary subdominant M(m7) chords are Mixolydian Augmented-fourth sonorities.

2. These are the chords actually generated by the harmonic series to partial eleven.

3. Secondary subdominant M(m7) chords are not considered dominant because they don't target degrees of Alpha Prime.

4. In context, if these chords target degrees of Alpha 6 or Beta Prime they may be considered dominant.

This is the way many paleo-theorists have done it in the past, but since Alpha Prime is, well, Alpha Prime, no overtone chord that doesn't come from a natural degree of Alpha Prime targeting a natural degree of Alpha Prime is a functional dominant harmony.

5. All of the altered dominant forms are available as subdominants as well.

6. The secondary dominant and secondary dominant sub-systems join to create a downward spiral of overtone sonorities.

7. That downward spiral of overtone chords covers the entire circa 144 semitone range of human hearing.

8. That pattern is imprinted in the subconscious of every human being, probably while they are still in the womb.

This why in absolute music - where music creates its own context - there has to be a musical contextual system or sub-system present that is independently functional. I'll present an example demonstrating this at the end of the first part of this book that describes the complete harmonic system. It really does sound like something that is deeply engrained, perhaps even at the genetic level.

*****

EXAMPLE 31:

Listen to Example 31

Of course, these examples can also transform directly.

OBSERVATIONS:

1. This is example twenty-eight with the harmonies transforming directly.

2. The continuous high surface tension provided by constant M(M7) chords is quite dissonant.

*****

EXAMPLE 32:

Listen to Example 32

OBSERVATIONS:

1. This is example twenty-nine with the harmonies transforming directly.

2. The increased surface tension provided by continuous m(m7) harmonies is much more mellow.

*****

EXAMPLE 33:

Listen to Example 33

OBSERVATIONS:

1. This is example thirty with the secondary overtone chords transforming directly.

2. While the M(M7) and m(m7) transformations were diatonic from chord to chord, these include chromaticism.

As the major thirds descend to minor sevenths in the target chords, as we also saw with direct transformations of secondary dominants.

3. The constant chromatic side-slipping creates a vaguely jazzy effect.

Though Mozart was no stranger to side-slips.

Why Music Works: Chapter Five

2010-09-20T03:27:00.004-05:00

The Harmonies of the Secondary Dominant Contextual Sub-System

PREFACE to All Posts:

This is to be the culmination of the Musical Relativity series of posts I did back in 2006, which can be found to your right in the sidebar. Back then I was calling the series Musical Implications of the Harmonic Overtone Series. Even before that, I did a series of posts called Harmonic Implications of the Overtone Series that started this all. Here, I am presenting the final weblog version of the evolving book I've decided to publish with the intention of getting some feedback before I create the final print version, which I plan to put into the ePub format for iBooks. So, please feel free to ask any questions about anything that you think I haven't made perfectly clear, and don't hesitate to offer any constructive criticisms or suggestions. Since this project is the accidental result of several decades of curios inquiry - and many prominent and also relatively anonymous theorists and teachers have contributed ideas to it (Which I will credit where memory serves and honor dictates) - I am eager to get a final layer of polish from any and all who may happen to read this series and find it useful, or potentially so. Since I am creating these posts as .txt files first, revision should be a simple process.

Since my pre-degree studies at The Guitar Institute of the Southwest and my undergraduate work at Berklee College of Music looked at music theory from the jazz perspective, and then my master of music and doctor of musical arts studies at Texas State and The University of North Texas were from the traditional perspective, a large part of how I discovered the things in this book-in-progress was the result of my trying to reconcile those different theoretical viewpoints. Since I want this work to be of practical value, I have retained all of the traditional theoretical nomenclature possible, and only added to it where necessary to describe phenomena that have not heretofore been present in musical analysis. I have, however, standardized terminology into what I think is the most logical system yet devised, and that will be explained as the reader goes along. There is a lot of built-in review and repetition - something I've learned from my decades of private teaching - so even a once-through with this systematic approach to understanding musical phenomena ought to be of significant benefit.

Outright addition to traditional musical analysis is limited to the symbology required to label root motion and transformation types so that the root motion and transformation patterns are visible: This greatly facilitates comprehension, and since good and bad harmonic continuities are separated by the effectiveness or lack thereof in the root motion and transformation patterns, this also actually functions as an aid to composition. All symbology - old and new - has been worked out over the past three decades so that everything is readily available with the standard letters, numbers, and symbols found on a QWERTY keyboard.

Finally, for the contextual systems, I have used the Greek alphabet: The normal diatonic systems - those comprised of two minor seconds and five major seconds - are Alpha, Beta, and Gamma. The exotic diatonic systems - those that contain a single augmented second - are Delta, Epsilon, and Zeta, and finally, the alien diatonic systems - those that contain two augmented seconds - are Eta, Theta, and Iota. Since the theoretical writings that started western art music out were handed down from ancient Greece, I thought this would be a fitting tribute, as well as a handy and logical classification scheme.

Basically, if you have a baccalaureate-level understanding of music theory from either a jazz or traditional perspective, you should have no problem understanding anything in this straight-forward treatise.

INTRODUCTION to Chapter Five:

In chapter one, I demonstrated how the overtone sonority generates the three normal diatonic systems - those seven note systems that contain two semitones and five tones: Alpha, Beta, and Gamma - and then in chapter two we examined each of those systems in detail, discovering that the primacy of Alpha is due to the fact that all seven of its harmonies can be arranged in progressive order. In chapter three, we examined the contextualization of Alpha Prime, looking at the various different root progressions types it can exhibit, their various transformations, and through this we also started to look into the world of musical effect and affect. Chapter four was dedicated to examining how Beta Prime and Gamma Prime compared to Alpha, using the same musical proof formats developed in chapter three. Through those proofs, we discovered some very unusual harmonic effects that evoke the uncanny that are contained in the Beta and Gamma systems.

Chapter five will take us out of the diatonic harmonic world and into the chromatic realm as we discover the origins of the secondary dominant sub-system sonorities.

*****

CHAPTER FIVE:

As the intuition of composers began to exhibit a more complete understanding of the overtone sonority's implications, they began to employ it for effect targeting degrees other than the tonic as secondary dominants. This process started with the nearest secondary dominants - V(m7)/V and V(m7)/IV - and progressed roughly a step at a time through V(m7)/ii and V(m7)/vi until finally the most remote secondary dominant, V(d5m7)/iii was reached. As we shall see, that V(d5m7)/iii - functioning first as V(d5m7)/V in the minor mode - unleashed entirely new classes of secondary dominant sonorities, many of which have not been properly described until I began to organize and classify them about five years ago (Though I figured out that the French Augmented Sixth was a V(d5m7)/V in minor when I was a doctoral candidate circa 1995).

EXAMPLE 20:

Listen to Example 20

To turn a I(M7) chord into a V(m7)/IV secondary dominant, all that is required is to lower the seventh by a semitone from a major seventh to a minor seventh, and to change a ii(m7) chord into a V(m7)/V secondary dominant, all that is required is to raise the third by a semitone from a minor third to a major third. This second formula - raising the third from minor to major - also works for V(m7)/ii and V(m7)/vi, but a funny thing happens if you apply it to the vii(d5m7) chord: You end up with a V(d5m7)/iii.

Though I have never been able to nail this down with any degree of certainty, since the common practice minor mode was based on Alpha 6, - the Aeolian mode, where the vii(d5m7) from Ionian is the ii(d5m7) chord - it seems most likely that the historical origins for this sonority were as a V(d5m7)/V - or v - in minor.

Since in the evolution of western art music counterpoint preceded harmony, the augmented sixth effect targeting the dominant degree that results from the major third above a diminished fifth was already known, so the orientation of the V(d5m7)/V when it first appeared was in second inversion. This is the so-called classic French Augmented Sixth sonority. Unfortunately, this historical baggage combined with a ridiculous and utterly non-descriptive name relegated the V(d5m7) sonority to the level of an obscure and difficult to understand curiosity. The same is true to an even greater degree with the so-called German Augmented Sixth sonority, because it actually has the intervallic structure of an overtone chord with enharmonic notation. As I shall demonstrate in this chapter, both of these sonorities are just altered secondary dominants, and their historically limited use is unfortunate and was unnecessary: Understanding this chapter will give any composer vastly increased sonic resources.

OBSERVATIONS:

1. This is example ten, which was example seven contextualized, with secondary dominants added.

All of these are the standard secondary dominants except for the V(d5m7)/iii. That chord could also be made into a standard secondary dominant by raising the fifth a half-step, but for the genesis of the other secondary dominant sonorities, I left it in its most natural state.

2. The V(d5m7)/iii is the point of origin for the altered dominant that generated Gamma Prime.

Remember, Gamma Prime is best described as a Phrygian mode with a major sixth and a major seventh, and Phrygian is the Alpha 3 mode here.

3. The V(d5m7)/iii is also the point of origin for the so-called "French Augmented Sixth" sonority.

4. The traditional so-called "French Augmented Sixth" chord is just a V(d5m7) in second inversion.

5. Historically, the V(d5m7)2nd probably first appeared in minor, where the ii(d5m7) became V(d5m7)2nd/V.

As you've certainly guessed by now if you've followed this series from the beginning, I have attempted to keep as much terminology and symbology from traditional classical and jazz theory as possible to make these concepts accessible to anybody trained in those disciplines, only modifying them and adding to them enough to properly describe the musical phenomena I'm defining. One set of modifications to the symbology is that I've replaced any arcane symbols with what can easily be found on a QWERTY keyboard, and another has been to at long last eliminate the figured bass formulations from inversions, as that old nomenclature is ponderous and confusing, even to me sometimes: It is much easier to understand a French sonority as V(d5m7)2nd - the 2nd meaning second inversion - than as a V(4/2/b).

6. The V(d5m7)2nd is just a naturally occuring altered dominant, available on any degree that can support a secondary dominant, and in any inversion.

We will see this in example twenty-one.

7. The diminished fifth in the V(d5m7) is an active leaning tone, replacing the passive perfect fifth in the overtone sonority.

8. Adding this additional active tone increases both the tension, and the resolution effect.

Once the intuition of composers lead them to discover that they could increase the resolutional desire of the overtone sonority by replacing a passive tone with another active tone, several new sonorities were created. In this case, the resolutional impetus is at 150% of normal, whereas later examples will completely double it.

9. To avoid parallel octaves, the root of a chord must always be a passive tone.

That is the case with the traditional French chord, but the upcoming fully-diminished seventh and German chords have no real root present in them, as we shall see.

*****

EXAMPLE 21:

Listen to Example 21

This is example twenty with secondary V(d5m7) on every degree until the final cadence, so as you can see, there are far more options for employing these chords than any of the traditional composers ever realized.

OBSERVATIONS:

1. This is example twenty with diminished fifths added to the secondary dominants.

2. A V(d5m7) can reside on any degree that can host a secondary dominant.

3. The traditional "French" terminology does not properly describe the function - or the origin - of these chords.

4. The traditional "French" terminology limits these chords to only one of four possible orientations, the second inversion.

5. The traditional "French" terminology is ridiculous, and must be abandoned.

*****

EXAMPLE 22:

Listen to Example 22

This is example twenty with minor ninths added to the secondary dominants.

Another set of secondary dominant function sonorities that are not taught properly are the so-called secondary fully-diminished seventh chords. What is usually called the root of these sonorities is actually a leading tone, so it's active and can't be the real root. The real root is always a major third below the leading tone, and so it's missing if all you are presented with is the symmetrical structure that consists of nothing but minor thirds (It's still OK to describe it as a fully-diminished seventh in that situation, so long as you realize that's just describing the structure, and not the function). The true function of the secondary fully-diminished seventh chord is as a secondary dominant with a minor ninth and no root: (Root), M3rd, P5th, m7th, and m9th. Since the root has to be a passive tone, the 9th in the transformational stratum is replacing the root with an active leaning tone. When looked at this way, the normal delayed crosswise transformation that secondary dominants make is not changed: 9 > 5, 5 > R, 7 > 3, and 3 > 7 after the resolutional interruption.

As with the V(d5m7) - the sonority formerly known as French - there are now three active tones in the upper stratum instead of two: The fifth is still passive... except for in the case of the V(d5m7m9)/iii that starts things off here. That sonority is the one traditionally described as a German Augmented Sixth, and all four voices in the transformational stratum there are active: (Root), M3rd (leading tone), d5th (leaning tone 1), m7th (leaning tone 2), and m9th (leaning tone 3).

OBSERVATIONS:

1. This is example twenty with minor ninths added to the secondary dominants.

2. The V(d5m7m9)/iii is the point of origin for the so-called "German Augmented Sixth" sonority.

3. The traditional so-called "German Augmented Sixth" sonority is just a V(d5m7m9) without a root, with the diminished fifth under the leading tone (To get the augmented sixth interval instead of the diminished third heard here), and the minor ninth in the bass (To lead into a second inversion sonority, and so avoid the parallel perfect fifths that result from the normal transformation of this chord, as we have here).

4. Historically, the V(d5m7m9/0) probably appeared first in minor, where the ii(d5m7) became V(d5m7m9/0)/V.

5. The V(d5m7m9/0) has the same intervallic structure as an overtone chord, except it is spelled enharmonically.

This has caused tons of confusion about the nature and function of this sonority. Basically, if you notate the D-sharp in the second measure above enharmonically as an E-flat, the transformational stratum is an F(m7)3rd chord. That coincidence - happy though it may be - has no bearing whatsoever on the functions of the notes in the chord: The F isn't a passive root, it's an active diminished fifth.

Nonetheless, generations of jazz musicians have been taught that errant way of looking at the chord through the so-called "substitute secondary dominant" theory: I know, because I was one of them. In that theory, to cite a single example, the V(m7)/I in C - a G(m7) sonority - can be replaced by a subV(m7)/I - which is a Db(m7) chord. Though expedient and simple - and certainly superior to the German terminology - this just isn't the way in which the overtone sonority implies that these chords are generated. The classical notation is correct, but its description is useless, while the jazz notation is incorrect, but at least its terminology is useful.

6. None of the notes in the traditional spelling can be the real root, however, because all of them are active.

7. The V(m7m9) chords also often appear without roots as so-called secondary fully-diminished seventh chords.

8. (If you're keeping up, you know this is not correct above: The notated third in a fully-diminished seventh is a passive 5th in the V(m7m9) chord from which it comes. - Geo) The notated root in a fully-diminished seventh chord is an active leading tone, so it can't be the real root.

9. A secondary fully-diminished seventh chord is properly understood as a V(m7m9/0): the root is simply missing.

10. Since real roots must be passive, minor ninths replace roots with active leaning tones.

11. With the minor ninth as a root substitute, the interrupted crosswise transformation is normal.

As with the V(d5m7) sonorities, the V(d5m7m9) chords can live on any degree that can carry a secondary dominant too.

*****

EXAMPLE 23:

Listen to Example 23

OBSERVATIONS:

1. This is example twenty with diminished fifths and minor ninths added to the secondary dominants.

2. A V(d5m7m9) can reside on any degree that can host a secondary dominant.

3. The traditional "German" nomenclature does not properly describe the function of these chords.

4. The traditional "German" nomenclature limits these chords to only one or two of four possible inversions.

5. The traditional "German" nomenclature is ridiculous, and must be abandoned.

6. Parallel perfect fifths result from the transformations of these chords: This is normal.

That is why there was often a so-called I(6/4) chord between the V(d5m7m9/0) and the tonic triad in common practice music; to avoid the parallel perfect fifth.

All of these secondary dominant types can also transform directly. While this maintains surface tension by always presenting a seventh chord, it sounds slippery and strange because the chromatically inflected leading tones are thwarted, and return to their non-inflected diatonic state instead of resolving. For me, it's an effect best used sparingly, but the rest of the examples in this chapter are 20-23 above with direct transformations.

*****

EXAMPLE 24:

Listen to Example 24

OBSERVATIONS:

1. This is example twenty with direct transformations of the secondary dominants.

2. Direct transformations are less natural sounding, as the leading tones are not resolved.

3. Direct transformations maintain greater surface tension, since a seventh chord is always sounding.

4. Choosing between directs or interrupts comes down to the effect/affect desired.

*****

EXAMPLE 25:

Listen to Example 25

OBSERVATIONS:

1. This is example twenty-one with the V(d5m7) chords transforming directly.

*****

EXAMPLE 26:

Listen to Example 26

OBSERVATIONS:

1. This is example twenty-two with the V(m7m9) chords transforming directly.

EXAMPLE 27:

Listen to Example 27

OBSERVATIONS:

1. This is example twenty-three with the V(d5m7m9) chords transforming directly.

I have managed to work all of the secondary dominant types into this systematic presentation with the exception of the so-called Augmented Seventh chords, where the passive perfect fifth is replaced by a minor sixth. I have that worked out, but not in this example set, so we'll now go on to secondary subdominants.

Why Music Works: Chapter Four

2010-09-17T07:27:00.003-05:00

Root Motion and Voice Transformation Types in the Beta and Gamma Contextual Systems

PREFACE to All Posts:

This is to be the culmination of the Musical Relativity series of posts I did back in 2006, which can be found to your right in the sidebar. Back then I was calling the series Musical Implications of the Harmonic Overtone Series. Even before that, I did a series of posts called Harmonic Implications of the Overtone Series that started this all. Here, I am presenting the final weblog version of the evolving book I've decided to publish with the intention of getting some feedback before I create the final print version, which I plan to put into the ePub format for iBooks. So, please feel free to ask any questions about anything that you think I haven't made perfectly clear, and don't hesitate to offer any constructive criticisms or suggestions. Since this project is the accidental result of several decades of curios inquiry - and many prominent and also relatively anonymous theorists and teachers have contributed ideas to it (Which I will credit where memory serves and honor dictates) - I am eager to get a final layer of polish from any and all who may happen to read this series and find it useful, or potentially so. Since I am creating these posts as .txt files first, revision should be a simple process.

Since my pre-degree studies at The Guitar Institute of the Southwest and my undergraduate work at Berklee College of Music looked at music theory from the jazz perspective, and then my master of music and doctor of musical arts studies at Texas State and The University of North Texas were from the traditional perspective, a large part of how I discovered the things in this book-in-progress was the result of my trying to reconcile those different theoretical viewpoints. Since I want this work to be of practical value, I have retained all of the traditional theoretical nomenclature possible, and only added to it where necessary to describe phenomena that have not heretofore been present in musical analysis. I have, however, standardized terminology into what I think is the most logical system yet devised, and that will be explained as the reader goes along. There is a lot of built-in review and repetition - something I've learned from my decades of private teaching - so even a once-through with this systematic approach to understanding musical phenomena ought to be of significant benefit.

Outright addition to traditional musical analysis is limited to the symbology required to label root motion and transformation types so that the root motion and transformation patterns are visible: This greatly facilitates comprehension, and since good and bad harmonic continuities are separated by the effectiveness or lack thereof in the root motion and transformation patterns, this also actually functions as an aid to composition. All symbology - old and new - has been worked out over the past three decades so that everything is readily available with the standard letters, numbers, and symbols found on a QWERTY keyboard.

Finally, for the contextual systems, I have used the Greek alphabet: The normal diatonic systems - those comprised of two minor seconds and five major seconds - are Alpha, Beta, and Gamma. The exotic diatonic systems - those that contain a single augmented second - are Delta, Epsilon, and Zeta, and finally, the alien diatonic systems - those that contain two augmented seconds - are Eta, Theta, and Iota. Since the theoretical writings that started western art music out were handed down from ancient Greece, I thought this would be a fitting tribute, as well as a handy and logical classification scheme.

Basically, if you have a baccalaureate-level understanding of music theory from either a jazz or traditional perspective, you should have no problem understanding anything in this straight-forward treatise.

INTRODUCTION to Chapter Four:

In chapter one, I demonstrated how the overtone sonority generates the three normal diatonic systems - those seven note systems that contain two semitones and five tones: Alpha, Beta, and Gamma - and then in chapter two we examined each of those systems in detail, discovering that the primacy of Alpha is due to the fact that all seven of its harmonies can be arranged in progressive order. In chapter three, we examined the contextualization of Alpha Prime, looking at the various different root progressions types it can exhibit, their various transformations, and through this we also started to look into the world of musical effect and affect.

Chapter four is dedicated to examining how Beta Prime and Gamma Prime compare to Alpha, using the same musical proof formats.

*****

CHAPTER FOUR:

EXAMPLE 14:

Listen to Example 14

Example fourteen is the same as example eleven, but with the inflections necessary to put it into the Beta Prime system. Obviously, these are end-contextualized, and are comparing the progressive resolutions to their opposite regressive versions.

The first of these root motions - the one from vii(d5m7) to bIII(A5M7) - is labelled PA5, for Progressive Augmented Fifth (This could also conceivably be labelled a Pd4 for Progressive Diminished Fourth, but since falling fifths are the most natural progressive root motions, I've decided to stick with fifths here). [As you'll see, my terminology eventually evolved into calling these quadra-tones. - Geo] This is a surprising and uncanny effect, because of both the root motion and the structure of the target harmony. Immediately following that - into the vi(d5m7) - we get a Progressive Tritone, which is labelled Ptt in the analysis. This root motion can actually be found in the Alpha system when moving from IV(M7) to vii(d5m7), but since the original intent of this example was to put all seven harmonies of Alpha into normal progressive order, we have not seen that yet.

After these initial histrionics - a very nice resource to affect the listener into the realm of the uncanny - the rest of the progressive relationships are relatively normal.

Since the second system is essentially the first system in reverse, the strangeness occurs near the end there. When I created this example back in 2008, I had still not completely nailed down the way I wanted to treat the progressive and regressive augmented fifths, so there is an Rd4 in the analysis, but the final version will say Rqt there for regressive quadra-tone. Note that the Ptt in Alpha is IV(M7) progressing to vii(d5m7) while the Rtt there is moving from vii(d5m7) to IV(M7): Both root motions are by tritone, but one is progressive and the other is regressive. The same thing applies in PA5 versus RA5.

OBSERVATIONS:

1. Overtone chord progressions imply falling fifths, hence, Progressive Augmented Fifth, Crosswise.

2. Overtone chord regressions imply rising fifths, hence, Regressive Augmented Fifth, Crosswise.

3. Progressive and regressive tritone root motions are perfectly usable in proper context.

4. Progressive and regressive augmented fifth root motions are perfectly usable in proper context.

5. Since augmented and diminished fifth root motions yield uncanny effects, employ accordingly.

6. The tonic minor/major seventh also evokes the uncanny.

7. The lowered mediant degree augmented/major seventh sonority also evokes the uncanny.

8. the harmonies and root motions possible in the Beta System provide new expressive resources.

*****

EXAMPLE 15:

Listen to Example 15

Example fifteen is the same as example twelve, but inflected to put it into Beta Prime.

OBSERVATIONS:

1. Beta system .5P's and .5R's contain less dramatically uncanny effects than the P's and R's did.

2. This is because the descending and ascending thirds smooth out the tritones and quadra-tones.

*****

EXAMPLE 16:

Listen to Example 16

This is example thirteen inflected to put it into Beta Prime

OBSERVATIONS:

1. Beta Prime SP's and SR's are again less uncanny than the system's P's and R's are.

In this case, all root motion and transformation is stepwise, hence the smoothness.

*****

EXAMPLE 17:

Listen to Example 17

Now we are ready to use example eleven/fourteen, example twelve/fifteen, and example thirteen/sixteen inflected into the Gamma System. By this point in creating these musical proofs, my terminology for the augmented fifth progressions and regressions had evolved to the point of referring to them as quadra-tones, which compares better to the tritone analysis symbols: Pqt and Rqt respectively, to fit in better with Ptt and Rtt. by this point, you should understand the analysis symbols well enough that the proofs become self-explanatory.

OBSERVATIONS:

1. Gamma System progressions and regressions contain even more uncanny sounding effects.

2. The areas of alternating quadra-tone and tritone root motions sound particularly sinister.

Obviously, these are gnarly sonic resources.

*****

EXAMPLE 18:

Listen to Example 18

Here are the half-progressions and half-regressions in Gamma. Note that I neglected to parenthetically denote the diminished thirds for the vii(d3d5m7) sonorities here. They are still notated properly - and so they sound correct - so that is just an oversight on my part.

OBSERVATIONS:

1. Gamma System .5P's and .5R's again contain fewer dramatically uncanny effects then the system's P's and R's do.

2. This is again, as before, because the .5P's and .5R's smooth out the tritones and quadra-tones.

*****

EXAMPLE 19:

Listen to Example 19

OBSERVATIONS:

1. Gamma Prime SP's and SR's are again less dramatically uncanny sounding than the system's P's and R's are.

2. Again, the fact that all root motion and transformation is stepwise aids smoothness.

3. Nonetheless, the Gamma System is filled with unsettling harmonic effects, which is an effective and affective resource.

Now that we have seen and heard the basic resources of the normal diatonic Alpha, Beta and Gamma contextual systems, it is time to enter the chromatic realm with secondary dominants and secondary subdominants as derived from the Alpha system.

Why Music Works: Chapter Three

2010-09-16T06:27:00.003-05:00

Root Motion and Voice Transformation Types in the Alpha Contextual System

PREFACE to All Posts:

This is to be the culmination of the Musical Relativity series of posts I did back in 2006, which can be found to your right in the sidebar. Back then I was calling the series Musical Implications of the Harmonic Overtone Series. Even before that, I did a series of posts called Harmonic Implications of the Overtone Series that started this all. Here, I am presenting the final weblog version of the evolving book I've decided to publish with the intention of getting some feedback before I create the final print version, which I plan to put into the ePub format for iBooks. So, please feel free to ask any questions about anything that you think I haven't made perfectly clear, and don't hesitate to offer any constructive criticisms or suggestions. Since this project is the accidental result of several decades of curios inquiry - and many prominent and also relatively anonymous theorists and teachers have contributed ideas to it (Which I will credit where memory serves and honor dictates) - I am eager to get a final layer of polish from any and all who may happen to read this series and find it useful, or potentially so. Since I am creating these posts as .txt files first, revision should be a simple process.

Since my pre-degree studies at The Guitar Institute of the Southwest and my undergraduate work at Berklee College of Music looked at music theory from the jazz perspective, and then my master of music and doctor of musical arts studies at Texas State and The University of North Texas were from the traditional perspective, a large part of how I discovered the things in this book-in-progress was the result of my trying to reconcile those different theoretical viewpoints. Since I want this work to be of practical value, I have retained all of the traditional theoretical nomenclature possible, and only added to it where necessary to describe phenomena that have not heretofore been present in musical analysis. I have, however, standardized terminology into what I think is the most logical system yet devised, and that will be explained as the reader goes along. There is a lot of built-in review and repetition - something I've learned from my decades of private teaching - so even a once-through with this systematic approach to understanding musical phenomena ought to be of significant benefit.

Outright addition to traditional musical analysis is limited to the symbology required to label root motion and transformation types so that the root motion and transformation patterns are visible: This greatly facilitates comprehension, and since good and bad harmonic continuities are separated by the effectiveness or lack thereof in the root motion and transformation patterns, this also actually functions as an aid to composition. All symbology - old and new - has been worked out over the past three decades so that everything is readily available with the standard letters, numbers, and symbols found on a QWERTY keyboard.

Finally, for the contextual systems, I have used the Greek alphabet: The normal diatonic systems - those comprised of two minor seconds and five major seconds - are Alpha, Beta, and Gamma. The exotic diatonic systems - those that contain a single augmented second - are Delta, Epsilon, and Zeta, and finally, the alien diatonic systems - those that contain two augmented seconds - are Eta, Theta, and Iota. Since the theoretical writings that started western art music out were handed down from ancient Greece, I thought this would be a fitting tribute, as well as a handy and logical classification scheme.

Basically, if you have a baccalaureate-level understanding of music theory from either a jazz or traditional perspective, you should have no problem understanding anything in this straight-forward treatise.

INTRODUCTION to Chapter Three:

In chapter one, I demonstrated how the overtone sonority generates the three normal diatonic systems - those seven note systems that contain two semitones and five tones: Alpha, Beta, and Gamma - and then in chapter two we examined each of those systems in detail, discovering that the primacy of Alpha is due to the fact that all seven of its harmonies can be arranged in progressive order.

Here we will look at the contextualization of Alpha Prime, the various different root progressions types it can exhibit, their various transformation types, and through this we will also begin to peer into the world of musical affect and effect.

*****

CHAPTER THREE:

EXAMPLE 10:

Listen to Example 10

Example ten is the contextualization of example seven, which was simply the ordering of the seven harmonies present in the Alpha System into a progressive order. To contextualize example seven into Alpha Prime, all that is required is to put a tonic harmony at the beginning, and a V(m7) P_ I at the end. The root motion form I(M7) to vii(d5m7) at the beginning - down by step - is called a Super-Regression, which gets the symbol SR in the analysis, and the voices transform in a clockwise - > - manner: R > 3 > 5 > 7 > R. I will present these in detail later in this chapter.

After that beginning contextualization, the harmonic continuity is just like example seven until measure eight, where the IV(M7) moves up into a V(m7) before the ending progressive resolution. The subdominant to dominant root motion - up by step - is the opposite of that at the beginning, and is called a Super-Progression. it gets SP in the analysis, and the voices transform in a counter-clockwise - < - manner: R > 7 > 5 > 3 > R. Opposite root motion types will always have opposite transformation types if they are circular and not crosswise.

Note also that Super-Progressions and Super-Regressions can produce parallel perfect fifths, as they do here into measure two, and into the final measure: This is normal. The biggest problem with traditional voice leading as it has been historically taught is that it is an amalgam of harmony and counterpoint, and not harmony isolated into its pure state, as we have here. In pure harmonic transformations, parallel perfect fifths sometimes result, and they are not a problem. In fact, it is the most organic and natural way of things, as the smooth transformational logic proves. Once you know this, the centuries of agonizing over whether to allow parallel perfect fifths in homophonic music becomes positively funny.

Another thing to notice is that in these root motions that are not Progressive, the transformations are direct, with no interruption as the Progressive root motions produce. The fact is, Progressive root motions can support direct transformations too, which I have demonstrated with the same continuity rendered that way on the second system: All of the Progressive transformations are direct until the final resolution to the tonic triad. Remembering that "P_+" is a Progressive Interrupted Crosswise Transformation, that becomes simply "P+" for Progressive Crosswise Transformation (The "P_" at the end is simply Progressive Interrupted Transformation, which is what you'll always see at perfect endings).

OBSERVATIONS:

1. In root motions other than progressive, transformations are direct, with no interruptions.

2. Progressive root motions may also support direct transformations.

3. The instant the overtone chord resolution varies, the realm of musical affect and effect is entered.

Affect and effect are nominally breached with all non-dominant progressive root motions that we've seen so far in the diatonic systems. One overtone sonority transforming to another through the chromatic system would be the default pure natural succession, and we'll look at that later. But really, when you allow for direct transformations over progressive root motions and especially non-progressive root motions, that is where the manifold harmonic effects that can effect the listener arise, and those are what we will be looking at for the rest of this chapter and the next.

4. Super-Regressions transform in a clockwise, circular manner: R > 3 > 5 > 7 > R.

5. Super-Progressions transform in a counterclockwise, circular manner: R > 7 > 5 > 3 > R.

6. Opposite root motions will always have opposite transformational directions, unless crosswise.

7. Direct transformations maintain surface tension by always presenting complete seventh chords.

This is one of the best features of direct transformations, as continuously interrupting the transformations sounds like a series of final-type resolutions, even if they are less perfect modal variants of the primary overtone sonority's resolution. Deft use of direct versus interrupted resolutions is one of a composer's basic resources for producing expressive effect.

8. musical contextualization is provided by beginnings and endings, or at least endings.

9. Omitting a contextualizing beginning can be an effective resource for affecting the listener.

Even in these examples, the beginning tonic harmony is a seventh chord, and so a stable context is not initially provided. In the following examples we will see how it is really the ending that provides a pure musical contextual definition. Obviously, starting the listener out in a foreign land, so to speak, and bringing them home can be a great musical plot device. This can be done by providing no initial context, or a false one.

10. Super-progressions and super-regressions can result in parallel perfect fifths, which is normal.

It is humorous, in retrospect, to view the conniptions some composers went to in order to avoid this effect in eras past.

*****

EXAMPLE 11:

Listen to Example 11

In these examples, I have only contextualized the endings to demonstrate observation nine above. Here, we also begin to look in detail at the types of root motion other than Progressive. The opposite of a progressive root motion is regressive, and it gets an "R" in the analysis (Many theorists have called these retrogressions in the past, but I prefer the simple yin/yang of two tri-syllabic words). This is what I have presented on the second system. There is also the root motion of a descending third into the penultimate measure of the second system, which is called a half-progression, and it gets .5P in the analysis. We'll see these in isolation in example twelve.

OBSERVATIONS:

1. Progressions move the transformational stratum lower.

2. Regressions move the transformational stratum higher.

3. Half-progressions and half-regressions result in three common tones between harmonies.

4. Progressions and regressions result in two common tones between harmonies.

5. Super-progressions and super-regressions result in one common tone between harmonies.

Composers need to know this, because the number of common tones between harmonies - note I'm referring to the transformational stratum - is what gives the effect of smoothness versus abruptness in the various root motion types.

6. The leading tone cannot be treated as a real root in a final resolution to a tonic triad.

7. The leading tone can be treated as a real root in an intermediate resolution to a tonic seventh chord.

If we were to attempt to move from the vii(d5m7) to I at the end of the second system, parallel octaves would result because both leading tones would move to the root of the tonic. While parallel fifths are fine in transformations that produce them in the upper stratum, parallel octaves are not between the two strata, for the simple reason that two voices are transforming the same; 7 > 1. Therefore, the leading tone cannot be treated as a real root in a final resolution (In an intermediate super-progression, of course, the upper stratum leading tone is held as a common tone, so it is 1 > 7). Discovering this was a much bigger deal than I initially thought, as it allowed me to figure out so-called secondary diminished seventh chords and also "German" augmented sixths, neither of which contain a real root, because all four tones are active: The root must be a passive tone.

8. Progressions (Not regressions as the observation mistakenly says) result in an incremental decrease of intensity, akin to musical gravity.

9. Regressions result in an incremental increase of intensity, akin to musical anti-gravity.

The phenomena of musical gravity and anti-gravity are quite real, as these examples demonstrate, and along with musical gravity comes a decrease of intensity as the pitch level of the transformational stratum lowers, while musical anti-gravity - or propulsion - brings with it an increase in perceived intensity as the pitch level of the transformational stratum rises. These are also effects the composer must be aware of.

10. Context can be nebulous or even missing at the beginning, so long as it is present at the end.

Again, unless an extra-musical context is provided, such as in a film score, where the scene creates the context.

*****

EXAMPLE 12:

Listen to Example 12

Now were ready to look at the isolated half-progressions and half-regressions. As you can see and hear, these root motion types are very smooth sounding due to all of the common tones involved.

OBSERVATIONS:

1. It takes two half-progressions to move the bass as far as one progression.

2. Two half-progressions transform the voices exactly the same as one progression.

3. It takes two half-regressions to move the bass as far as one regression.

4. Two half-regressions transform the voices exactly the same as one regression.

I did not give the root motion types arbitrary names. Rather, starting with the resolution of the overtone sonority as a normative progression, I compared all other types to it, and named them logically: The opposite of a progression is a regression, so half of a progression or a regression is exactly that.

5. Half-progressions are musical gravity moving at half speed.

6. Half-regressions are musical anti-gravity moving at half speed.

7. Direct octaves occur between the bass and a transforming voice in half-progressions: This is normal.

Half-regressions do not have this feature, because the bass moves into a voice that is tied in the upper stratum. This is important to note, as in a half-progression, the root of each new chord is a new note that did not exist in the previous harmony, while in a half-regression the bass is not a new note, but one already established in the previous harmony. This is one of the features that produces the different effects between the two root motion types.

8. Half-progressions transform clockwise, and half-regressions transform counter-clockwise.

Missing above, but I'm going to recreate these in Sibelius anyway for the final version, I think.

*****

EXAMPLE 13:

Listen to Example 13

Finally for this chapter, here are the super-progressions and super-regressions. Despite the stepwise smoothness of the bass, these root motion types sound quite abrupt because there is only a single common tone between the adjacent harmonies.

OBSERVATIONS:

1. Two super-progressions move the bass as far as three progressions.

2. Two super-progressions transform the voices down as far as three progressions.

3. Two super-regressions move the bass as far as three regressions.

4. Two super-regressions transform the voices up as far as three regressions.

5. Super-progressions are musical gravity moving down at 1.5 times normal speed.

6. Super-regressions are musical anti-gravity moving up at 1.5 times normal speed.

For 5 & 6, this despite the direction of the bass line.

7. Super-progressive transformations can result in parallel perfect fifths: This is normal.

8. Super-regressive transformations can result in parallel perfect fifths: This is normal.

9. Super progressions transform counterclockwise, super-regressions transform clockwise.

We have now seen all of the root motion types in the Alpha Prime system with the exception of the tritone root motion that occurs between IV(M7) and vii(d5m7). This root motion type will be encountered in Beta Prime and Gamma Prime - where they are unavoidable if we follow the pattern of the examples presented in this chapter - so we will examine those, and more, in chapter four.

Why Music Works: Chapter Two

2010-09-14T15:27:00.003-05:00

Analysis of the Modal Sub-Contexts of the Alpha, Beta, and Gamma Systems

PREFACE to All Posts:

This is to be the culmination of the Musical Relativity series of posts I did back in 2006, which can be found to your right in the sidebar. Back then I was calling the series Musical Implications of the Harmonic Overtone Series. Even before that, I did a series of posts called Harmonic Implications of the Overtone Series that started this all. Here, I am presenting the final weblog version of the evolving book I've decided to publish with the intention of getting some feedback before I create the final print version, which I plan to put into the ePub format for iBooks. So, please feel free to ask any questions about anything that you think I haven't made perfectly clear, and don't hesitate to offer any constructive criticisms or suggestions. Since this project is the accidental result of several decades of curios inquiry - and many prominent and also relatively anonymous theorists and teachers have contributed ideas to it (Which I will credit where memory serves and honor dictates) - I am eager to get a final layer of polish from any and all who may happen to read this series and find it useful, or potentially so. Since I am creating these posts as .txt files first, revision should be a simple process.

Since my pre-degree studies at The Guitar Institute of the Southwest and my undergraduate work at Berklee College of Music looked at music theory from the jazz perspective, and then my master of music and doctor of musical arts studies at Texas State and The University of North Texas were from the traditional perspective, a large part of how I discovered the things in this book-in-progress was the result of my trying to reconcile those different theoretical viewpoints. Since I want this work to be of practical value, I have retained all of the traditional theoretical nomenclature possible, and only added to it where necessary to describe phenomena that have not heretofore been present in musical analysis. I have, however, standardized terminology into what I think is the most logical system yet devised, and that will be explained as the reader goes along. There is a lot of built-in review and repetition - something I've learned from my decades of private teaching - so even a once-through with this systematic approach to understanding musical phenomena ought to be of significant benefit.

Outright addition to traditional musical analysis is limited to the symbology required to label root motion and transformation types so that the root motion and transformation patterns are visible: This greatly facilitates comprehension, and since good and bad harmonic continuities are separated by the effectiveness or lack thereof in the root motion and transformation patterns, this also actually functions as an aid to composition. All symbology - old and new - has been worked out over the past three decades so that everything is readily available with the standard letters, numbers, and symbols found on a QWERTY keyboard.

Finally, for the contextual systems, I have used the Greek alphabet: The normal diatonic systems - those comprised of two minor seconds and five major seconds - are Alpha, Beta, and Gamma. The exotic diatonic systems - those that contain a single augmented second - are Delta, Epsilon, and Zeta, and finally, the alien diatonic systems - those that contain two augmented seconds - are Eta, Theta, and Iota. Since the theoretical writings that started western art music out were handed down from ancient Greece, I thought this would be a fitting tribute, as well as a handy and logical classification scheme.

Basically, if you have a baccalaureate-level understanding of music theory from either a jazz or traditional perspective, you should have no problem understanding anything in this straight-forward treatise.

INTRODUCTION to Chapter Two:

In chapter one, we looked at the structure of the harmonic series and found that it is what theorists call, for good reason, a dominant seventh chord: A major triad with a minor seventh. After removing the fundamental generator and its superfluous perfect twelfth, I identified the basic musical forces that reside within the tritone - the leading tone and leaning tone impetuses - and demonstrated how these forces imbue the overtone sonority with a desire for resolution. We found that with the five part texture of pure harmony, the resolution creates a delayed or interrupted crosswise transformation in the upper stratum, and with a single further diatonic resolution the Alpha Prime contextual system is created. Then we saw how with a resolution to a minor triad the Beta Prime diatonic contextual system results, and finally, how by starting from a V(d5m7) - the point of origin for the ridiculously so-called French Augmented Sixth sonority - that the Gamma Prime contextual system results.

Here in chapter two, we will look at the independent and dependent sub-systems of Alpha, Beta, and Gamma. Furthermore, I will demonstrate the primacy of Alpha Prime, the comparatively limited nature of Beta, and the outright flawed character of Gamma.

*****

CHAPTER TWO:

EXAMPLE 7A:

On the top system, I have presented all of the harmonies of Alpha Prime in standard notation. Underneath, I have put the analysis symbols in the format I will be using. A capital Roman numeral will represent a major triad, and a small case Roman numeral will be assumed to be a minor triad, unless there is a further indication for the fifth, as is the case for vii(d5m7) in the final measure, where the fifth is diminished. The parenthetical indicators are for alterations to the triad, descriptions of the seventh, and also any upper structure tones that may be present: M, m, A, d represent Major, minor, Augmented, and diminished, respectively. This results in more detail than is often presented: For example, what is usually called a V7 in most other systems is a V(m7) here. I prefer this more perfect logic and level of detail. Finally, under the chord analysis symbols is the tone/semitone pattern for Alpha Prime: 2, 2, 1, 2, 2, 2, 1. Obviously, this is the traditional major mode specifically called Ionian.

When I was coming up, diagrams almost precisely like example 7a were given at or near the beginning of harmonic theory. This really does a disservice to the student, as it can lead to the incorrect notion that scales generate harmony, when the truth - as I demonstrated in chapter one - is the other way around: Harmonic continuities generate the scales or modes.

With this in mind, I have demonstrated how the displacement modes of Alpha Prime are best represented by their less perfect resolutional paradigms that imitate the original: Alpha independent sub-context 2: Dorian is defined by v(m7), i(m7), and IV(m7); Alpha independent sub-context 3: Phrygian is defined by v(d5m7), i(m7), and iv(m7); Alpha independent sub-context 4: Lydian is defined by V(M7), I(M7), and #iv(d5m7); Alpha independent sub-context 5: Mixolydian is defined by v(m7), I(m7), and IV(M7); Alpha independent sub-context 6: Aeolian is defined by v(m7), i(m7), and iv(m7); and finally, the Alpha dependent sub-context 7: Locrian is defined by bV(M7), i(d5m7), and iv(m7). The Locrian mode is the only dependent sub-context in the Alpha system because it does not contain a perfect fifth, and so it has no proper dominant function harmony, and the i(d5m7) is not stable enough to provide a proper conclusion. In a chromatic versus a diatonic context, it is perfectly possible to target a (d5m7) harmony with an overtone sonority - or any other a perfect fifth above - so it is primarily the dissonant nature of the sonority that renders it unable to function on its own. Within a larger independent context, of course, Locrian effects are a perfectly fine resource.

EXAMPLE 7B:

Listen to Example 7B

The Alpha System allows for all seven harmonies to be put into a progressive order. As we shall see later, the construction of the Beta and Gamma systems do not allow for this. Here is one factor that explains the primacy of Alpha. Additionally, all of the diatonic degrees from vii(d5m7) up to and including the tonic can carry secondary dominant harmonies, as I shall demonstrate later. Furthermore, from the primary subdominant of IV(M7) on, the secondary subdominants can continue the cycle into the chromatic realm. Ultimately, a barber pole loop consisting of the primary subdominant and secondary subdominants leading away from the tonic into the chromatic realm will prepare for the most remote of the secondary dominants leading back to the primary dominant and then the tonic. That will be the ultimate musical proof, whereas this is the initial.

OBSERVATIONS:

1. All master contexts will have functional dominant, tonic, and subdominant harmonies.

2. All independent sub-contexts will have functional dominant and tonic harmonies.

3. A non-functional subdominant harmony does not destroy the independence of a sub-contextual mode.

4. A dominant harmony with a diminished fifth is functional.

5. Only one sub-contextual mode of the Alpha System is contextually dependent: The Locrian mode.

6. A string of harmonies in progressive order creates a double harmonic canon in the upper stratum.

This isn't glaringly obvious without some elaboration of the parts, but - within the upper stratum - the soprano voice follows the tenor by a measure at the fourth above, and the alto follows the bass at the same distance and interval. This is such a cool phenomenon, that I'll devote an entire future chapter to it.

7. A string of harmonies in progressive order lowers the two strata.

As you can see, as the voices progressively transform, they get lower. This is what I call musical gravity, and it is the same phenomenon that Heinrich Schenker discovered, but he never figured out exactly what he was looking at: He was seeing an artifact of music that has a preponderance of progressive harmonic relationships in it. Since progressive root motions are statistically the most common type in western art music, you'll get the 3, 2, 1's; 5, 4, 3, 2, 1's &c. It's no big deal, really, and I can't think of too many musical exercises more futile than Schenkerian analysis: While mildly interesting, it's laborious and doesn't really teach the composer anything of much practical value.

The Natural Laws of Pure Harmony:

A slightly elaborated list. I'm still figuring out how best to list, word, and present these observations.

1. Pure harmony consists of five total voices.

2. These five voices are divided into a four-part, close-position transformational stratum above a constant-root bass part.

3. All chords are in root position in pure harmony.

4. The upper stratum consists of complete seventh chords, or triads with a doubled root.

5. The upper stratum transforms in a crosswise or circular manner, depending upon the root motion type.

Number five is a bit of a peek ahead, as I will show the other root motion types at a later point.

*****

EXAMPLE 8A:

This example of the Beta Contextual System is in the same format I presented for Alpha. The top system is a simple lineal presentation of the harmonies, and then the six displacement modes are on the following three systems. Alpha Prime - the Ionian mode - is considered the model for comparison, so any deviations from the pattern established there are represented in the analysis symbols. For example, the third degree of Alpha Prime is major, so the harmony residing on the minor third degree of Beta is described as a bIII(A5M7): the small case "b" standing in for the flat symbol. This convention will be followed throughout.

The proper way to name Beta Prime is as a Dorian mode with a major seventh. Independent sub-context 2, therefore, is described as a Phrygian mode with a major sixth. The third sub-context in Beta, Lydian augmented fifth, is a dependent sub-context, because the augmented triad cannot function as a tonic, and therefore there is also no dominant or progressive relationship between #v(d5m7) and the tonic.

Beta 4 is again an independent sub-context, and it is properly described as a Mixolydian mode with an augmented fourth. As I mentioned previously, this is the actual scale created by the harmonic series to P11.

Because the tonic harmony for Beta 5 is an overtone sonority, it also has to be compared to the Mixolydian mode, which means it is a Mixolydian with a minor sixth. Likewise, Beta 6 & 7 have (d5m7) chords on the tonic degree, so they are best compared to the Locrian mode: Locrian major second in the case of Beta 5, and Locrian diminished fourth in the case of Beta 6: Of course, both Beta 5 & 6 are dependent sub-contexts.

EXAMPLE 8B:

Listen to Example 8B

NOTE: I had to use separate example scores, so yes, the first harmony in 8B starts with a triad moving into a seventh. I just noticed that, sorry.

Whereas all seven harmonies in Alpha could be arranged in a progressive order, here in Beta only five of them line up that way.

OBSERVATIONS:

1. An overtone chord can be a functional subdominant.

2. A minor.major seventh can be a functional dominant.

3. The mode created by the harmonic series to P11 would not allow the overtone chord to function as a dominant.

This means that, as far as music is concerned, the harmonic series is complete at P7.

4. The Beta System has only three independent sub-contexts.

5. Fully three Beta System sub-contexts are dependent on outside contextual definition.

6. The primacy of the Alpha System is demonstrated by the fact that all seven of its harmonies can be ordered in progressive relationships.

7. Traditional so-called melodic minor is a bi-modal combination of Alpha 6 and Beta Prime.

*****

EXAMPLE 9A:

Here we have the Gamma contextual system, which is the last of the normal diatonic systems generated by the harmonic system, normal being defined as systems consisting of two semitones, and five tones. Gamma Prime is best described as a Phrygian mode with a major sixth and major seventh since the minor second degree is what distinguishes Phrygian from the other minor modes in Alpha. The Gamma system is interesting and difficult to navigate because only one of its sub-contexts, Gamma 4, is independent: All the rest are dependent. Gamma 7 is particularly bizarre, because the tonic triad has a diminished third, a the rest of the scale contains diminished fourth, and a diminished fifth.

EXAMPLE 9B

Listen to Example 9B

Due to this structure, only three of the harmonies in the Gamma system can be arranged in a progressive order.

OBSERVATIONS:

1. The Gamma System has only one independent sub-context.

2. Fully five of the Gamma System's sub-contexts are dependent on outside contextual definition.

3. Only three Gamma System harmonies occur in progressive order.

4. Due to 1-3, the Gamma system is the antithesis of the Alpha system, and the Beta system lies in between.

5. The Gamma System ammounts to a hexatonic whole tone scale with one of the tones filled in.

6. The augmented fifth/minor seventh on bIII - commonly called an augmented seventh chord - is a byproduct of the genesis of the Gamma System.

7. Both of the altered dominants on bIII and V can be used in non-diatonic contexts.

More on this later.

8. Between the Alpha, Beta, and Gamma Systems, all normal diatonic resources are present.

9. The combined diatonic contextual resources available - independent and dependent - totals 21 modes.

10. Twelve of these modes are independent, while 9 are contextually dependent.

11. Three additional contextual systems are possible allowing for one augmented second.

I've worked these out, and will present them later.

12. A further three contextual systems are possible allowing for two augmented seconds.

I thought I had worked these out, but I can't find them, so I may be mistaken here. In any event, this is a topic for much later. The next chapter will be looking at and listening to the various root motions in the Alpha system in context.

*****

Why Music Works: Chapter One

2010-09-12T15:27:00.004-05:00

How the Harmonic Series Generates the Three Diatonic Contextual Systems: Alpha, Beta, and Gamma

PREFACE to All Posts:

This is to be the culmination of the Musical Relativity series of posts I did back in 2006, which can be found to your right in the sidebar. Back then I was calling the series Musical Implications of the Harmonic Overtone Series. Even before that, I did a series of posts called Harmonic Implications of the Overtone Series that started this all. Here, I am presenting the final weblog version of the evolving book I've decided to publish with the intention of getting some feedback before I create the final print version, which I plan to put into the ePub format for iBooks. So, please feel free to ask any questions about anything that you think I haven't made perfectly clear, and don't hesitate to offer any constructive criticisms or suggestions. Since this project is the accidental result of several decades of curios inquiry - and many prominent and also relatively anonymous theorists and teachers have contributed ideas to it (Which I will credit where memory serves and honor dictates) - I am eager to get a final layer of polish from any and all who may happen to read this series and find it useful, or potentially so. Since I am creating these posts as .txt files first, revision should be a simple process.

Since my pre-degree studies at The Guitar Institute of the Southwest and my undergraduate work at Berklee College of Music looked at music theory from the jazz perspective, and then my master of music and doctor of musical arts studies at Texas State and The University of North Texas were from the traditional perspective, a large part of how I discovered the things in this book-in-progress was the result of my trying to reconcile those different theoretical viewpoints. Since I want this work to be of practical value, I have retained all of the traditional theoretical nomenclature possible, and only added to it where necessary to describe phenomena that have not heretofore been present in musical analysis. I have, however, standardized terminology into what I think is the most logical system yet devised, and that will be explained as the reader goes along. There is a lot of built-in review and repetition - something I've learned from my decades of private teaching - so even a once-through with this systematic approach to understanding musical phenomena ought to be of significant benefit.

Outright addition to traditional musical analysis is limited to the symbology required to label root motion and transformation types so that the root motion and transformation patterns are visible: This greatly facilitates comprehension, and since good and bad harmonic continuities are separated by the effectiveness or lack thereof in the root motion and transformation patterns, this also actually functions as an aid to composition. All symbology - old and new - has been worked out over the past three decades so that everything is readily available with the standard letters, numbers, and symbols found on a QWERTY keyboard.

Finally, for the contextual systems, I have used the Greek alphabet: The normal diatonic systems - those comprised of two minor seconds and five major seconds - are Alpha, Beta, and Gamma. The exotic diatonic systems - those that contain a single augmented second - are Delta, Epsilon, and Zeta, and finally, the alien diatonic systems - those that contain two augmented seconds - are Eta, Theta, and Iota. Since the theoretical writings that started western art music out were handed down from ancient Greece, I thought this would be a fitting tribute, as well as a handy and logical classification scheme.

Basically, if you have a baccalaureate-level understanding of music theory from either a jazz or traditional perspective, you should have no problem understanding anything in this straight-forward treatise.

INTRODUCTION to Chapter One:

My foremost intention with this monograph is to, at long last, describe in a comprehensive way the basic musical forces that exist, and to present them and their resultants in a way that will be of use to composers. These forces all spring from a tension that is inherent in the harmonic series, which desires a resolution. Allowing or thwarting this desire for resolution is what allows music to express things.

While many twentieth-century theorists attempted to describe this through mathematics, their efforts failed because of one simple oversight: The overwhelming majority of possessors of musical minds are not mathematically inclined. Our minds generally work in terms of shapes and sounds: If we can visualize a thing or hear a thing in our minds, we can freely manipulate it to our heart's content. Numbers don't look or sound like anything to us, so they might as well not exist in our world (With the exception of proportions and draw-out geometry). Eventually, this made me return to the beginnings of scientific thought, which was actually called natural philosophy. Sir Isaac Newton described himself as a natural philosopher. Music theory has never even reached the Newtonian stage of evolution because the fundamental forces have never been defined and demonstrated in a rigorous and logical way.

A natural philosopher looks at the God-given constant in a thing, and extrapolates its implications out to describe the observable phenomena (Or hearable phenomena, in this case). For music, that constant is the harmonic overtone series, and so I have used musical proofs based on the harmonic series instead of mathematical proofs to demonstrate all of the resolution possibilities present, as well as all of the manifold contextual results that it can produce.

*****

CHAPTER ONE:

The overtone series is a harmonic system, so music is a harmonic system. If we look at the evolution of western art music, the harmonic series underpinnings were known from ancient Greek writings, but the implications of those writings were only discovered bit by bit by starting at the beginning, which was the unaccompanied melody of Gregorian plainchant. From plainchant, an admittedly oversimplified synopsis would proceed to organum, fauxbourdon, polyphony, traditional homophony, and then jazz, which was the real end result in the twentieth century: The seventh chords were finally ubiquitously used as tonics, including the overtone sonority itself in blues music.

Where the problem arose with both so-called traditional music theory and the succeeding jazz music theory was that the elements were never separated into their pure states: Pure harmony and pure counterpoint. By the time Joseph Schillinger properly described pure harmony, the "classical" guys had jumped the tracks and gone off into the desert of desiccation and musical nihilism known as - to cite but a single representative monicker - atonality (Though admittedly, some good did come of this for film music, but that's not what they intended). Meanwhile, even the most prominent Schillinger student of all, George Gershwin, never really exhibited in his works the revelation that was pure harmony: In other words, like all jazz cats, he didn't give a rip about the transformational voice leading that Schillinger revealed.

Finally, Schillinger himself never coupled his pure harmony with the overtone sonority, so he missed a crucial link, and that's why many of his ideas stray into the weeds of pure speculation. Admittedly, The System was a hodge-podge put together in haste by some of his students after his sudden and untimely death, so there is the possibility - regardless of how remote - that he did in fact make this connection. In any case, I made the connection independently due to the accidents of fortunate circumstance combined with my natural curiosity, and so here we are.

As mentioned previously, the natural philosopher's approach is to begin at the beginning, which in the case of music is with the harmonic overtone series, that every natural sound carries with it (A computer generated pure sine wave, which is an artificial phenomenon, would be an exception to this). In nature, the recognizable timbre of a sound is defined by the attack transients, formants, amplitudes, envelopes, and the phase relationships between the partials in the overtone series, which are individually pure sinusoidal periodicities.

Unfortunately, the pure waveform soundfont I found does not work in iTunes for some reason - it works fine in Encore though - so we're stuck with an organ sound that has relatively few harmonics present in it. If you play the overtone series in 12√2 equal temperament, this is what you get (I will address the simple logic of twelve-tone equal temperament much later in this book).

EXAMPLE 1:

Listen to Example 1

Though TTET does not render the mathematically perfect ratios of overtones on a vibrating string or in a vibrating column of air - which themselves vary minutely, especially on a string - it is quite close enough: A mathematically perfect overtone series sounds virtually the same, as many years of programming digital synthesizers taught me. To put a finer point on it, this 12√2 rendering of the series sounds like a so-called dominant seventh chord, as did the perfect ratio renderings I've created in years past with instruments such as the Synclavier. So please, spare me the, "TTET destroyed tonality" nonsense: The ratios in TTET may be irrational, but the concept isn't - 1/3 is an irrational number too, but a child can understand the idea of a third part of something. Likewise, 1/12 of an octave is a perfectly rational concept, even if the resulting ratios are expressed by irrational numbers.

Under the system are the various eras of western art music development - Plainchant, Organum, Fauxbourdon, Polyphony, Homophony, and Jazz - which roughly correspond to composers coming to understand ever higher partials in the harmonic series. That this understanding was primarily intuitive and not analytical need not concern us, as in musical composition - and performance - it has usually been intuition that lead the way, with theorists describing practice later.

Next I have labelled the partials 1-8, which I will call P1 through P8. It is useful to draw a distinction between partials and harmonics, as the fundamental generator of the series is not a harmonic: Only the overtones P2 and on are harmonics. By labeling the partials starting with the fundamental as P1, the series works out for us the pure intervallic ratios, which are found on the third, fourth and fifth lines.

Though self explanatory, 2:1 is the ratio for the inviolable perfect octave, 3:2 is the ratio for the just perfect fifth, 4:3 is a just perfect fourth, 5:4 is a just major third, 6:5 is the large minor third, 7:6 is the small minor third, and 8:7 is the first of the just major seconds. Line four shows the ratios of 5:3 for the major sixth and 8:5 for the minor sixth, and finally we have 7:4 for the minor seventh and 7:5 for the tritone on the bottom line.

OBSERVATIONS:

1. All adjacent intervals and their inversions from P1 through P7 are consonances.

2. All adjacent intervals and their inversions beyond P7 are dissonances.

Neither of these two observations should require any comment.

3. Consonances that remain super-particular ratios when inverted are considered perfect.

4. Consonances that are not super-particular ratios when inverted are considered imperfect.

A super-particular ratio is a ratio in which the terms differ by 1: All adjacent ratios for the consonances from P1 to P7 are super-particular; 2 - 1= 1, 3 - 2= 1, &c. Only those consonant intervals that remain super-particular when inverted are considered perfect, however: The 2:1 octave is unchanged during an inversion - which technically requires two octaves of displacement to avoid the unison - and the 3:2 perfect fifth inverts to a 4:3 perfect fourth, which are both super-particular: 3 - 2= 1 and 4 - 3= 1.

With the major and minor thirds, however, we get 8:5 for the minor sixth and 5:3 for the major sixth, which are not super-particular: 8 - 5= 3 and 5 - 3= 2.

5. The only non-adjacent dissonance with a single skip within the first seven partials is the tritone.

6. The only other non-adjacent dissonance within the first seven partials is the minor seventh itself.

The ratios of 3:1, 4:2, 5:3, and 6:4 are all consonances: Perfect twelfth, perfect octave, major sixth, and perfect fifth. Only 7:5 is a dissonance, it being the tritone, and then 8:6 is another perfect fourth. The minor seventh requires two harmonics to be skipped from 7:4, and it's the only other dissonance from P1 to P7 (And of course, it converts to the 8:7 dissonant major second).

7. The harmonic series is complete at P7; P8 is only present to yield the minor sixth harmonic ratio.

As I said at the beginning of this chapter, "The overtone series is a harmonic system, so music is a harmonic system." Once the adjacent intervals in the series become dissonances at 8:7, the harmonic part of the series is over and the melodic part of the series begins. Sure, inversions of harmonic structures can yield seconds, but a root position harmony in close position will only contain adjacent thirds. While ninths can replace doubled roots in the basic harmonic musical system - and sixths can replace fifths - elevenths and thirteenths are actually acquired from harmony with two transformational strata, which is a subject for much, much later (Transformational stratum one is Root, 3rd, 5th, and 7th, while transformational stratum two is 9th, 11th, 13th, and Root: A ponderous system outside of the basic harmonic nature of music, but one useful for certain effects within a larger musical or extra-musical context). Theorists who claim that twentieth century music was an exploration of the harmonic series beyond P7 are in error.

8. Since the harmonic series contains a dissonant tritone, it is inherently unstable.

The life forces of music are in the tritone - the leading tone and leaning tone impetuses - and the inherent instability of the overtone sonority - a dominant seventh, remember - is what provides music with its possibility for forward motion. This is one reason that wild-eyed criticisms of 12√2 temperament are misguided: Since the harmonic series is inherently dissonant, why would a stable just tuning be an advantage, or even particularly desirable? Since just ratios are only possible with computers, voices and other variable pitch instruments like strings, it is just silly to criticize TTET, which is a perfectly equitable solution for fixed pitch idioms. Later, we will look at just how small the deviations from just are with TTET.

*****

The first thing that the harmonic series from P1 to P7 does for us is to define the format for pure harmony: Since we are dealing with a harmonic system, all we have to do is eliminate the non-harmonic fundamental generator and the superfluous twelfth that it produces to get this. The resulting structure is Root, Root, third, fifth, and seventh, which is the pattern for pure harmony.

1. Pure harmony consists of five total voices.

2. These five voices are divided into a four-part, close position transformational stratum above a constant-root bass part.

3. Only the root is doubled - or trebled after a resolution - in pure harmony.

4. Though the transformational stratum can be in any close position inversion, all harmonies are root position in pure harmony due to the constant-root bass part.

*****

EXAMPLE 2:

Listen to Example 2

The four different notes of the overtone sonority are divided into two pairs: The root and perfect fifth are passive tones, while the tritone involving the major third and the minor seventh consists of the two active tones.

OBSERVATIONS:

1. The root is the foundation of the overtone sonority.

2. Together with the root, the perfect fifth provides context for the dissonant tritone.

Since there are twelve pitch classes in the chromatic system and the tritone involves two of them, there are only six tritones possible. Since there are twelve possible overtone sonorities, this means that each tritone is shared between two possible roots. In this case, if the tritone between B-natural and F-natural is enharmonically notated as C-flat to F-natural, it can belong to the overtone sonority with the root on D-flat and the perfect fifth of A-flat. That means that the two possible roots for each tritone are also in a tritone relationship with each other, as well as the two possible perfect fifths. When you change the context of a tritone with the other perfect fifth a tritone away, you also reverse the functions of the notes involved in the tritone. In this case, the B-natural leading tone becomes a C-flat leaning tone, and the F-natural leaning tone becomes an F-natural leading tone.

This is how jazz theorists justify their concept of substitute secondary dominant harmonies, by the way, though we will see later that this isn't really valid according to the implications of the series.

3. The root and perfect fifth are passive tones, neither desiring to rise or fall.

4. The major third and minor seventh are active tones, desiring to resolve their shared dissonance.

5. The major third is a leading tone, and it desires to rise by a semitone.

6. The minor seventh is a leaning tone, and it desires to fall by either a semitone or a tone.

7. The perfect fifth may rise or fall by a tone as the tritone resolves to a major target.

8. The root may remain stationary or fall a perfect fifth when the tritone resolves.

9. In order for the target sonority to be in root position, the lower root must fall by a perfect fifth.

10. In order for the target sonority to be complete, the upper root must remain stationary.

11. In order to avoid doubling a potential active tone in the target sonority, the perfect fifth must fall by a tone.

*****

When we follow the above observations, the following primordial resolution of the overtone chord is produced.

EXAMPLE 3:

Listen to Example 3

This is how you say, "The End" in music: The overtone sonority with a doubled root resolves to a targeted major triad with a trebled root. If we were to wish a continuation, the unison C-natural in the transformational stratum would have to dissolve with one of the C's going down to a seventh - either B-natural or B-flat.

The falling perfect fifth/rising perfect fourth root motion is called a Progressive root motion, and it will get a capital P in the analysis. When we get to more elaborate examples, they will not be called, "chord progressions" because a progression is this specific type of root motion; rather they will be referred to as harmonic continuities.

Though interrupted by the doubled root in the upper stratum, if one of the C's moved down into a seventh, this would be what is called an interrupted or delayed crosswise transformation, as you can see from the diagram in between the staves: The root and fifth exchange functions, and the seventh and third also exchange functions after the third's resolution into the unison. This would be a P_+ in the analysis, which reads, Progressive _Interrupted +Crosswise transformation. Uninterrupted crosswise transformations, though less than perfectly natural, can also be used when that effect is desired, as I shall demonstrate later.

OBSERVATIONS:

1. This resolution does not yield all seven tones of the diatonic system.

2. This resolution - or similar less perfect versions - does, however, yield all six tones of the ancient hexaphonic Church modes.

The modern Ionian mode implied here was not really a common Church mode, but as we shall see, modal displacements of this formula will produce modes commonly used back then. It is also useful to note that when ending resolutions first appeared, they were primitive versions of this one.

3. In order for the target chord to become a complete seventh chord, the doubled root in the transformational stratum must fall.

4. The doubled root may fall either a semitone or a tone.

5. If the doubled root falls by a semitone, the potential for a diatonic system will be possible.

6. If the doubled root falls by a tone, the potential for a diatonic system will be destroyed.

7. If the doubled root falls by a tone, another overtone sonority will be created.

The rule for creating diatonic systems through this resolutional paradigm is to retain the inflection of the notes present in the preceding chords. Here, for example, one would have the doubled root descend to B-natural, because that note is present in the preceding dominant harmony. Allowing for chromaticism with this paradigm will lead to various integrated modalities; again, we'll see this at a later point.

8. As the diagram shows, 1 becomes 5, 5 becomes 1, 7 becomes 3, and 3 becomes 7 after the interruption of the resolution.

9. Therefore, progressive resolution of the overtone sonority creates an interrupted crosswise transformation.

10. A single additional progressive resolution would complete a diatonic system.

*****

If we allow for that dissolution of the unison C so that the former major third descends to B-natural - becoming a major seventh - and add an additional progressive resolution, the Alpha Contextual System is produced.

EXAMPLE 4:

Listen to Example 4

The Alpha Contextual System has as Alpha Prime the traditional major or Ionian mode. Within this contextual system are the additional sub-contexts known as the displacement modes of Dorian, Phrygian, Lydian, Mixolydian, Aeolian, and Locrian.

OBSERVATIONS:

1. With two progressive resolutions from the overtone sonority, the diatonic system is complete.

The only note missing in the single cycle resolution was A-natural, which now appears as the major third of the subdominant harmony.

2. All three possible harmonic functions - dominant, tonic and subdominant - are also now defined.

3. The progressive resolution from the tonic to the subdominant is a less perfect form of progressive resolution.

4. Additional less-than-perfect progressive resolutions will be found in the modal sub-contexts.

5. There exists an additional diatonic Beta Contextual System with resolution to a minor triad.

6. The Alpha Prime tonic scale is 2, 2, 1, 2, 2, 2, 1: The semitones are separated by two tones.

*****

If we start again with the formula and have the initial resolution to a minor tonic, the Beta Contextual System is produced.

EXAMPLE 5:

Listen to Example 5

The Beta Prime mode is like a Dorian mode from the Alpha System with a raised seventh degree. In common practice minor key music, this system was combined with the Aeolian mode to produce the nonatonic so-called melodic minor scale: Roughly speaking, his is the ascending version of that system, and Aeolian was used as the descending form (Though those conventions weren't always adhered to).

OBSERVATIONS:

1. The tonic seventh chord in this system is a highly dissonant minor triad with a major seventh.

This means the third mode here has an augmented triad, of course.

2. The subdominant chord in this system is another overtone sonority.

3. The sub-contextual subdominant mode is the scale that the overtone series creates to P11.

The overtone scale is best described as a Mixolydian mode with a raised fourth degree, not, "Lydian flat seven" as some jazz theorists describe it: The home harmony is a dominant seventh, not a major seventh.

4. There exists an additional Gamma Contextual System with resolution from a dominant harmony containing a diminished fifth.

The point of origin for the V(d5m7) chord - and the so called French augmented sixth - is from the V/V in minor, where the chord on the second degree is a ii(d5m7) before the third is raised to make it a secondary dominant. In Alpha Prime that sonority appears as the remote V(d5m7)/iii harmony. I will demonstrate this when we get to the secondary dominant harmonies, but here I'm just demonstrating the three possible diatonic contextual systems that contain two semitones and five tones.

EXAMPLE 6:

Listen to Example 6

The Gamma Prime scale created through this resolution process is best described as a Phrygian mode with the sixth and seventh degrees raised, since the minor second degree is the distinguishing characteristic of the Phrygian mode.

OBSERVATIONS:

1. Diminishing the fifth of the dominant chord makes that fifth into an active leaning tone.

As I just mentioned, this V(d5m7) has a natural origin in the Alpha System. When the fifth was D-natural, it was a passive tone that could theoretically rise or fall during the resolution, so long as the composer is prepared to deal with doubling a potential active tone. Now, the D-flat is a third active tone in the dominant harmony that desires to resolve down by semitone to the new root. This additional impetus increases the resolutional desire of the dominant harmony.

2. The tonic seventh chord is again a highly dissonant minor/major seventh.

3. The subdominant chord is again another overtone sonority.

4. The Gamma Prime tonic scale is, 1, 2, 2, 2, 2, 2, 1: The semitones are not separated.

5. All 21 possible diatonic modes consisting of five tones and two semitones have now been generated.

I will present these in detail in chapter two.

*****

What lead me to the idea of pure musical contextual systems was a phenomenon I noticed with so-called atonal works: They were completely unsatisfying - unlistenable, actually - in a purely musical concert context, but when used in a stage play or a film score, they could become quite effective. What I finally realized is that the play or the film provided an extra-musical context in which these pieces could be effective. Likewise, episodes of atonality within a larger purely musical context that is based on any one of the 21 diatonic modes that are independent sub-contexts - those with major or minor triads as tonics, and not diminished or augmented triads - can also be effective. When music provides its own context, it has to be based on an independent musical contextual system or sub-system, regardless of any arguments to the contrary.

Imitation Study Number 4 in A Minor

2010-09-01T00:27:00.000-05:00

This is an interesting piece. It's a fugue with the answer on the subdominant and in inversion, and the initial statement of the subject has an accompaniment line. This was necessary because the subject begins on the third degree of the mode, which may very well be unique. I've never encountered another subject like this, anyway.

Interestingly - to me - this is actually the very first fugue subject I ever came up with back in 1986 or 1987. I got the idea while studying through Joseph Schillinger's System of Musical Composition. Historically, fugue subjects have begun on the tonic or dominant degrees - or with approach notes to those degrees - and I wondered why not the mediant, since it is a degree of the tonic triad? So, I came up with this wedge shaped theme with the idea of writing a two-part invention for guitar with it. That idea fizzled because I just had no idea how to proceed.

Later, when I was working on my masters degree, I wrote a three-part invention for string trio with the subject, which I really, really like. In that piece, I came up with the chromatic countersubjects and the solution of having the answer on the subdominant degree and in melodic inversion.

So, here we are 23 years later and I've come full circle back to the idea of writing a two-voice solo guitar piece on the subject; only now I have the compositional technique to make it work.

Here's today's MIDI to M4A conversion:

Imitation Study Number 4 in A Minor

Despite the subject's unusual feature of starting on the mediant degree of the minor mode and having a diverging pair of melodic trajectories, it's actually fairly Bachian (Since Bach was my inspiration here). However, you'll notice the G-sharp a the beginning of measure two is followed by a G-natural one beat later: This is strange, and it sets up a tension that I really like a lot. Furthermore, on the last eighth note of measure two, there is a diminished octave - equalling a major seventh in sound - between the C-sharp in the bass and the C-natural in the subject, and the final sixteenth is a diminished seventh (Which equals a major sixth in sound, of course). These bold cross-relations give the piece the dark feel of a lament: Something is just not right with the universe, and so there is quite a bit of expressive sadness to the piece.

This is only reenforced by the true counter-answer in measure three, which is an ascending chromatic line at the beginning, and a descending chromatic line at the end. Pathos. On the last eighth of measure four, we have a diminished fourth (plus an octave) and then a diminished tenth (Which is the inversion of an augmented sixth). This is another highly unusual feature of the exposition.

As I mentioned in the previous post, many times two voice fugues work out better with a thrid statement of the thematic unit, so that two subjects appear and you can reveal the true countersubject: That's the case here. Just as the answer is inverted from the subject, the counter-answer is inverted for the countersubject. IMO, this is very, very cool.

Most of the time I use tail figures for the episodes in my fugues, but in this case the head works better. Since there are losts of wide leaps, this also allows for range adjustments that keep the elements within an executable range on the guitar. I made that adjustment in measure eight, which set up the upcoming middle entries, which are in the relative major.

The major mode version of the subject and answer worked out better starting off on the tonic degree of the triad, and the countersubject and counter-answer therefore ended up diatonic instead of chromatic. this provides a happy sounding contrast to the dark sounding exposition.

The second episode that starts at fourteen required another range adjustment in fifteen, whereas the string trio version allowed the violin to continue into the stratosphere.

The second middle entries are all about displaying the contrapuntal inversions of the previously revealed material. the original orientation of answer and counter-answer are first stated, only in the dominant level minor of E, and then the countersubject over subject appears starting in nineteen.

Episode three at twenty-one takes the piece to the subdominant minor region, where I present the answer over counter-answer orientation. At the end of this, we get an augmented sixth, of course. Both of these inverted orientations sound very dark. Darker than one might expect just judging from an intervallic analysis. I've never really been able to figure out why this is - even the consonances sound dissonant - but I really like the effect (Even if it was a happy - or sad, I guess - accident).

The final page gets us to the concluding stretto section - also the first stretto I ever composed - and there is no episode needed to get here because the subject to the previous answer is back in A minor. Is that cool, or what?

The first 1.5 beat overlap appears in measure twenty-seven, and then the 2.5 beat dovetail starts in twenty-eight. To get everything back in sync vis-a-vis the bar line, I relax that back to 1.5 beats in the first half of thirty, and then just the head appears in the second half, which sets up the conclusion.

Obviously, it's bedtime.

Imitation Study Number 3 in F-sharp Minor

2010-08-13T23:27:00.000-05:00

Here's the third entry in the series, but I must mention that the sequence of these in the final set of pieces will surely change. That will depend on the keys, time signatures, and types of pieces these end up being. This one happens to be an actual two-voice fugue with a tonal answer at the fifth, as opposed to the previous two, which were two-part inventions, per Bach's definition, with real answers at the octave.

Number three here also ended up in the tres cool key of F-sharp minor, whereas the previous two were in the boring and overworked key of A minor. This being a collection for solo guitar, at this point I'm thinking that A minor pieces will probably end up alternating with those in other keys... if I end up with enough pieces in other keys. lol.

Since these are solo guitar pieces written in open score, they can also function as duets for students too, which is a nice feature.

The subject of today's fugue was created one day while I was driving from San Antonio to San Marcos when I was a masters student at Texas State. I'm not positive anymore, but I think it was 1989 or 1990. It just came to me while I was daydreaming on the drive, and I sang it out in solfeggio.

First it became a rather lengthy and undisciplined string trio piece, but later it became a very tight fugato for chamber orchestra. I'm sure I'll still use that fugato in a larger piece at some point, but it's nice to have a solo - and duet - guitar version of it.

This is not simply a transcription of the fugato, though: I had to do quite a bit of re-composing to get this guitar version... or perhaps rearranging would be more accurate: The orchestral interjections are gone, naturally, but some of the episodes are redone as well.

Here's today's MIDI to AAC conversion:

Imitation Study Number 3 in F-sharp Minor

As you can see if you read the previous posts in this series, this isn't the stately Musical Offering/Art of Fugue type of subject I used for those pieces. This is a sprightly type of subject with lots of leaps and tied notes that nonetheless gradually accelerates to become a head/tail type of deal.

One of the coolest things about the answer is the counter-answer that works with it: There's a descending chromatic tetrachord at the beginning, and the ascending version at the end. I remember being amazed when I discovered this... and, well, it is kind of amazing. There's an eighth rest at the beginning of the last beat of measure 6 to avoid the unison on G-sharp, but it remains implied and a duet performance could actually execute it. Oh, and I have the high notes in the head of the subject and answer notated as harmonics, but it is technically possible to execute them as normally attacked notes.

If you are writing a two-part fugue, many times - but not always - the exposition will work out better with a third statement so you end up with the subject twice: That's the case here.

Note also the even more amazing feature of the countersubject's beginning: It's yet another ascending chromatic tetrachord!

The first episode, such as it is, is just a single measure spin-out of the subject's tail. One of the things I got better at here was register shifting: To get this kind of music to work on the guitar, it's necessary to shift registers to keep the music within the guitar's quite limited contrapuntal range. The descending minor seventh from F-sharp to G-sharp makes the required adjustment smoothly.

By not chromatically inflecting the sixteenths on the final beat as la and ti, they become fa and sol to effect the modulation to the relative key of A major. The low E's in this section are why the piece ended up in F-sharp minor.

I needed a new version of the countersubject at 11 because the descending chromatic tetrachord wouldn't work there, so it's a simple diatonic accompaniment. Same deal at 14, as an ascending chromatic tetrachord wouldn't work there either, so I used another diatonic version of the countersubject. I had to make another range concession at 15, but it sounds very natural; as if it's a natural part of the piece. Bach was a genius at this, but I'm just now getting really good with it.

Note that these are both subject statements and not answers.

Measure 17 is another single measure episode, and I used another octave displacement to set up the final statement in the relative...

... which is finally the answer, complete with descending and ascending chromatic tetrachords in the counter-answer. This sounds really awesome in the major mode, as you usually don't get this sort of thing in that context. I like it a lot. Love it, even.

The episode at 21 is now two measures: 21 modulates back to the tonic minor with the introduction of the raised submediant and leading tone in the final four sixteenths - a new figure for the fugue - and then 22 allows for the required register adjustment. Note that measure 22 is just like the first episode that modulated the piece to the relative, except that the raised sixth and seventh degrees are retained on the last beat, so there is no modulation.

The recapitulation at 23 is a super-close stretto between the subject in the bass and the tonal answer above: Only a single beat of delay. This is an extraordinarily difficult thing to pull off, and I really don't know how I make these things happen; I just "notice" them for some reason. Remember, I just sang this subject while on a boring drive up the interstate: I had no idea the chromaticism would work, or that this stretto was possible at the time. I didn't even work out the tonal answer until later.

I've never had a student who could do this, unfortunately, and in fact, I've never met or heard of anyone else who can do it either (Nobody living, anyway). It must be an inexplicable and rare gift, I guess. Too bad it's not worth any money in today's world. lol.

Notice that I had to make a register adjustment even in the subject to get this to work on the guitar there in 25: One hardly notices it. I hadn't acquired this skill when I came up with this subject twenty years ago, and that's part of the development of the craft aspect of contrapuntal writing: It's nice to have the gift of natural talent, but only time and effort can give you mastery of the details of the craftwork.

The final episode beginning at 26 has a gnarly contrary motion dovetail between the tail of the subject and the tail of the answer; it moves into major seconds and out to octaves. In the orchestral version, these are ninths and double-octaves, which are smoother sounding, but his is really excellent on solo guitar (Or, between two guitars playing a duet). the straight sixteenths that originally appeared back in 21 no reappear in 27, and the piece resolves out to a six-note F-sharp minor chord at 28. That's it: 28 measures and about a minute and a half. Very tight and efficient.

I'm sure I've used this image before, but she's just so striking and natural.

Imitation Study Number 2 in A Minor

2010-08-04T20:27:00.001-05:00

Well, this series has found a name now: Imitation Studies. I'm not calling them inventions because, 1] They do not all answer at the octave - some of these are fugues, IOW - and 2] Bach already used that. lol. Believe it or not, I've completed five of them so far, and the previous one was "perfected" by simply changing the dotted-eighth/sixteenth rhythm of the episodes to quarter/eighth triplets. This works better because the triplet feel comes right out of the tail of the subject. Since that was such a minor change, I decided to move on to the next in the series.

With this piece, however, the dotted-eighth/sixteenth rhythm is in the subject, while the written-out trill is in the countersubject: Therefore the episodes sound more natural and organic with the dotted-eighth/sixteenth figuration of the subject. Nice discovery.

I composed this subject back when I was a doctoral candidate at UNT in the early 90's, and it was the final project for the graduate level Invertible Counterpoint and Fugue course I took there. Since I composed it as a four-part canonic stretto with one measure delays, it took the form of an epic string quartet piece (You can also see the score in PDF format if you want).

Well, while writing this guitar invention version, I discovered some new facets of the subject, so that got me to thinking about revising the fugue. I think I'm coming up with a new compositional process here: Guitar invention or two-voice fugue first, then execute the final fugue for whatever ensemble turns out to be most appropriate. Bonus is, I get a bunch of guitar pieces out of all my ensemble fugues. Go me!

Number 2 here also ended up in A minor, but two of the next three are in the "cool" keys of B minor and F-sharp minor (Number 6 I'm working on is in E major).

Here's todays MIDI to M4A conversion:

Imitation Study Number 2 in A Minor

This is a five measure head, body, and tail subject, so we get the desirable odd bar length and an accelerating rhythm until the point of resistance in four and the cadential figure at the end. The countersubject is straight quarters until the final measure, and then there is a dotted-quarter/eighth followed by the written-out trill (I've come up with several of these written-out trills now, and they are becoming a thematic sub-language for me).

For the main episode I chose a descending chromatic line... hey, I like those.

... and this first iteration is non-modulatory.

Since the subject works as a four-part canon, the organizational scheme is of ever closer answers. Starting in 17 we hear the countersubject above the subject for the first time, but this is interrupted in 21 by the subject overlapping itself by one measure. Then, the rest of the countersubject seamlessly continues out of the subject's cadential figure due to the way I constructed it to work like this (I rock).

In 26 the former episode reappears, but this time it's foreshortened by a measure and modulates to the dominant minor region. Sorry I crammed it onto a single system, but it was a matter of keeping the page count down for this post version.

The sweeping rising thirds in the bass starting at 31.5 - me, sol, ti, re, fa, le - sound wicked pisser, as we used to say at Berklee, and then we get another mesure of the countersubject before the next interruption with now two measures of overlap.

For my esthetic, the third episode has to be different - with Bach everything is beautifully, maddeningly, and organically different - and here I discovered that the subject works over the descending chromatic figure of the episode. This leads to some unusual contrapuntal motions, the coolest of which is the perfect fifth into a diminished fifth from 41 into 42. I need to put this in the string quartet, because it will be even more effective with three or four voices. At the end I modified the tail of the subject and the bass part to set up the modulation to the relative major region. Hot, or not?

We're down to two measures of delay/up to three measures of overlap now, and I changed the second measure of the countersubject in 45 just because it sounds really beautiful this way (Hey, I do have a heart). Same with the bass part in 49 (Mahler said, "Interesting is easy; beautiful is hard"). I thrive on interesting, but I go for beautiful whenever I can.

By the way, the inversion of this stretto would not work, because parallel fifths would be implied. The implied elevenths sound fine (I tried it the other way for grins... but those turned to frowns).

The original episode then reappears at 51, but yet another measure shorter as the piece returns to the tonic for the final stretto... or so it would appear.

The trill figure takes the final canon from five measures to six, and the end is a bit underwhelming, which sets up the hyper-stretto coda: Subject over subject in rhythmic augmentation. This is admittedly a bit weird in two voices, as there is a leapt-into minor ninth at the beginning of 64. Since the implied harmony is so obviously a V(m7m9) though - and this is immediately strengthened by the appearance of the major third - it actually works, IMO. Bach would never have done this (Well, he never did, to my knowledge), but I think it's kind of cool.

I worked up to a nice flourish in 71 and 72 with an implied V(4/2)/iv and then an augmented sixth into the primary dominant. A final - and new - trill figure finishes things off with a final chromatic flourish.

As it stands now, this is a better composition than the original fugue... which is why I have to re-write that puppy.

No doubt about that being hot.

Invention in A Minor v5.3

2010-07-23T22:27:00.000-05:00

This piece is now "finished." I put the quotation marks there because pieces like this tend to get revised over succeeding years - sometimes radically - but it is complete as it is, and I'm quite happy with it. One of the things I like about composing with dispassionate and lifeless notation based MIDI sequencers is that if I like the way it sounds in MIDI, I know a well performed version will be even better.

Here's the m4a sound file:

Invention in A Minor

This is actually version 5.3, as you can see, so it really didn't take all that many versions to get it done.

There are no changes here, so if you want a rundown, see the previous post.

The changes start at the first stretto section in the subdominant minor region. Previously I had both the 2.5 measure delay and the 1.5 measure delay stretti here, but I decided to save the closer one for the upcoming dominant minor. This was the breakthrough idea that allowed me to complete the piece, because I hadn't saved anything for the dominant previously.

There are no changes to the episode starting at 17 or the relative major statement starting at 20 either.

This episode starting at 23, however, is entirely new. The inversion of the major mode statement does not work due to a leapt-into major second - which is an augmented second in the minor version - so this was a natural place to put the harmonized subject in the bass with a melody in the lead. Sounds cool in MIDI, but it would be a PITA to play. Since this is primarily a compositional exercise, though, that's of little practical consequence.

At 27 we get the dominant minor statement with the closer stretto, and this also gives the piece a melodic climax at the C in 28. By saving this for the peak, we get a dramatic pause under it as the next subject statement begins, which sounds nice tres cool. Since the piece is 48 measures long now, 48/28.5= 1.684, which is just about as close as one can get to the Golden Mean of 1.618. Nice, huh?

Saving this stretto for this point also makes it out of kilter vis-a-vis the bar lines, so at the conclusion of the section in 31 I was able to use another nice contrapuntal/rhythmic acceleration lick into the final episode: quarter-eighth, quarter-eighth triplets, and then dotted quarter-sixteenth, dotted quarter-sixteenth. I like this effect, and in performance I'd probably swing the dotted quarter-sixteenth sections anyway - so it doesn't sound so stiff - so the transition is super-smooth.

From the final episode that begins at 32 until the end is the same as the previous version...

... so if you are new to this series and have any questions, check out the previous post.

Now that this has got me exploring some aspects of rhythmic variation in counterpoint - something I'd like to develop more as it's has heretofore been one of my weaker points - I believe I'll write two-voice inventions with all of my previously composed subjects. Series work like this is a great way to develop compositional technique, and after my eighteen axial studies and twenty-four figuration preludes, a series of inventions would be a logical step.

Invention in A Minor v4.5

2010-07-20T23:27:00.000-05:00

At first I thought I was working on a two-voice fugue with this subject, then it became an invention, and now it's actually almost a piece of music. Previously I had the various statements and stretti linked only by lineal licks, but now I've put some real episodes into it.

Here's today's AAC sound file:

Invention in A Minor v4.5

The exposition hasn't changed any, but beginning in measure seven there is an episode that modulates to the subdominant minor region. This is actually a harmonized version of the subject, which you can see if you look at the first dotted-eighth note of every group: It's the note at the corresponding location in the subject until I prepare to modulate at the end. Since the tonic A minor is the dominant degree of the upcoming D minor region, this was easy to accomplish by using the written-out trill on the proper level and introducing the C-sharp leading tone at the end.

An interesting aspect of working in two voices is that you can't easily just work your rhythmic velocity up to a constant eighth note motoric motion - or alternating dotted-eighths and sixteenths, as the case would be here - so that makes one think about how to make the combined rhythm breathe smoothly. This actually requires a bit of thought. Fortunately, the subject has six different rhythmic values in it and the trill tends to scrub out any buildup and so allows for a new beginning, so to speak. That's why I put the trill figure at the end of the episode: Going back into a constant quarter cumulative rhythm sounds excellent with that trill as a re-transition.

Since I've now established that there will be no perpetual motion cumulative rhythm, the stretti beginning in eleven breathe quite naturally from a rhythmic standpoint.

This culminates in a virtual stop in the rhythm at fifteen, which sounds quite dramatically cool at the new pitch climax of B-flat.

There is another episode based on the harmonized subject beginning at eighteen, but I've modified the counterpoint to facilitate a modulation from the subdominant minor region to the relative major. This took a little more head scratching and chin rubbing than the more obvious and easy previous episode, but the result is more interesting, which is what you want in succeeding versions of a musical idea.

The relative major statement is the last confirmed bit of the piece from the beginning - I'm sure the first 23 measures won't change any - but I had to resort to a lineal link to get to the final episode based on the harmonized subject, which is back in the tonic minor. I'm just not sure what will replace measure 24 at this point, but I'm thinking of statements of some kind on the dominant minor level, which would provide an overall pitch climax on C above the previous high of B-flat. I must admit to being stuck here, but that's usually a good sign, as what comes after much consideration is always - in my experience, at least - the final element that makes the piece perfect (Or pluperfect, if I'm particularly inspired).

For the final version of the harmonized subject episode, I discovered that two consecutive statements of that harmony, with slight variation, would work with a descending chromatic tetrachord as the counterpoint. this is goosebump-inducing cool if the listener is astute enough to follow what's going on, and it's one of those musical effects that can only be perceived with a bit of retrospection. So, 25-27 have the first statement over do, ti, and te...

... while the second statement is over la, le, and sol. It isn't really possible to perceive the new beginning of the subject until half-way through 28, so the effect is quite surprising. Tres cool, non?

There is no change to the closest stretto that begins at 31 - which functions as a recapitulation here - but the hyper-stretto coda that starts at 35 has been improved with some very nifty chromatic, rhythmic, and contrapuntal action.

The last quarter of 38 now has an augmented sixth on it - the only one of those in the piece at this point - and then there is an ascending chromatic tetrachord in 39. Then, the descending quarter triplet that starts in 40 has some cool contrapuntal relationships over the rhythmically augmented trill: m7-M6, m6-P5, and d5-m6 into the m6-m7 and d5 last two-beat triplet. It sounds a bit awkward in the m4a version because MIDI assumes every note is attacked, but when played the trill will be executed with hammer-ons and pull-offs, so it should be very smooth and cool. It's a nice closing contrapuntal device.

Almost more interesting to me though is the cumulative rhythm produced. If you consider a half note as the basic rhythmic unit, starting in 39 the cumulative attacks are 2, 4, 6, 3, 1, which is a very nice accelerating and decelerating rhythmic ratio.

I'm learning some cool things about the musico-psychological effects of rhythm with this piece.

I haven't posted a redhead in a while.